Asymptotically quasiconformal four manifolds

T.Kato

Introduction

For topological manifolds, there are several kinds of additional structures. Of particular interest is the inclusion top ⊂ quasiconformal ⊂ smooth, since the situation drastically changes according to the dimension. One of the most important work was done by Donaldson and Sullivan ([DS]) where an example of topological four manifold was given which cannot admit any quasi conformal structure. On the other hand the equality top = quasi conformal holds for dimension larger than 5 ([S]). Donaldson’s polynomial Q(M) is defined over symmetric power of H_{2}(M : Z) which is smooth invariant. In [DS], Q is constructed over quasiconformal four manifolds. In particular it turns out that it is invariant under quasiconformal homeomorphisms. Notice any smooth manifold admits a unique quasiconformal structure compatible with the smooth one.

For a four manifold M, let us take an open subset R ⊂ M and a smooth disk D, where closure R̄ = clR ∈ M is homeomorphic to D. In order to compare smooth structures on M̌ = M\R̄ and M̂ = M\D, between smooth open four manifolds, we will introduce a notion, asymptotically quasiconformal homeomorphism between these. It is consisted by a family of quasiconformal mappings. In this paper we will use gauge theory as an analytic tool for the family of maps. Like homology theory, we will reconstruct ASD moduli spaces by using underlying Banach spaces which are much larger than W_{k}^{2} Hilbert spaces. Two basic constructions are as follows; we construct ASD moduli spaces over cylindrical manifolds where we use uniformly bounded L^{2+ϵ} or L^{4+2ϵ} function spaces as the underlying Banach spaces. Then we will formulate an asymptotic ASD connection, which is consisted by a family of approximately ASD connections. It turns out that the moduli space of asymptotic ASD connections contains the ordinary ASD moduli space. If it could bubble at infinity, then this new moduli space also contains such element. This extended moduli spaces will be used to compare two cylindrical manifolds which are asymptotically quasiconformally homeomorphic.

Let D, D′ ⊂ R^{4} be a compact domain with Riemannian metrics g and g′ respectively. Then we say that a homeomorphism f : (D, g) ≅ (D′, g′) is quasiconformal and has constant bounded by K if the following numbers are pointwisely bounded:

H(φ)(x) = lim Sup_{r→0} \frac{max\{|φ(y) − φ(x)| : |x − y| = r\}}{min\{|φ(y) − φ(x)| : |x − y| = r\}} < K, φ = f, f^{−1}.

The equality holds H(f)(x) = exp(d([g], [f^{∗}(g′)])) where [ ] denotes the conformal class (4.A). Thus H(f)(x) measures distance at x , between one metric and its pull back by the quasiconformal homeomorphism. Usually one uses the cylindrical metric g on M̂, namely g|end is isometric to g|S^{3} +dt^{2}, where g|S^{3} is a fixed smooth metric on S^{3}. In relation with the construction of gauge theory passing through quasiconformal mappings, we will benefit from the Gehring’s theorem ([G]) which states; if I : (K, g) ≅ (K′, g′) is a quasiconformal homeomorphism, then ∇I ∈ L^{4+δ} and δ > 0 is essentially determined by ess supH(I). Notice that the pull backed Riemannian metric I^{∗}(g′) can be approximated by a smooth family g_{i} in L^{2} (lemma 4.2). If we deonte I_{i} = id: (D, g_{i}) ≅ (D, I^{∗}(g′)), then ess supH(I_{i}) are uniformly bounded. In particular H(I_{i})^{2} − 1 → 0 in all L^{n}. ¿From these properties, we will introduce the following notion, asymptotically quasiconformal structure. Let us equip a complete Riemannain metric g of bounded geometry on M̂.

An asymptotically quasiconformal homeomorphism (ACH) between M̌ and M̂ consists of the following data:

(1) exaustion by compact subsets K_{0} ⊂⊂ K_{1} ⊂⊂ ⋯ ⊂ M̌,

(2) a family of Riemannian metrics (K_{l}, g_{l}) of uniformly bounded geometry, (|g_{l} − g_{l′}|K_{l}|C^{∞} are uniform),

(3) quasi conformal embeddings I_{l} : (K_{l}, g_{l}) ↪ (M̂, g)

so that these data satisfies the following:

(4) for any x ∈ M̂, there exists l(x) ≥ 0 such that image of I_{l} contains x for all l ≥ l(x), and I_{l}(m) → ∞ as m → ∞,

(5) for all m ∈ M̌, d(I_{l}(m), I_{l′}(m)) → 0 as l, l′ → ∞,

(6) |H(I_{l}^{±1})^{2} − 1|L_{loc}^{N} → 0 for a sufficiently large N >> 0,

(7) |∇(I_{l′}^{±1} − I_{l}^{±1})|L_{loc}^{4+δ}(K_{l}) → 0 for l′ ≥ l and some δ > 0.

It is known that there exists an open subset R′ ⊂ R^{4}, homeomorphic to R^{4} such that the induced smooth structure on R′ is different from R^{4} ([DF]). Then one notices the following; there exists R′, an exotic R^{4} which admits an exaustion K_{0} ⊂ K_{1} ⊂ ⋯ ⊂ R′ and a family of C^{∞} embeddings I_{l} : K_{l} ↪ R^{4} satisfying (4) and (5) above.

In the following theorem, one may take M = ♯_{l}X, X = CP^{2} or S^{2} × S^{2} for some l ≥ 1.

Theorem 0.1 There exists a pair (M, R) as above, such that M̌ ≡ M\R̄ cannot be asymptotically quasiconformally homeomorphic to M̂ = (M\D, g).

In particular the pair (M, R) has the property that any homeomorphism between (M̌, h) and (M̂, g) cannot be locally boundedly conformal with respect to their smooth Riemannian structures.

Using the result in [DS] (following the argument in [FU] p20), one may have an example of R ⊂ M = 3(S^{3} × S^{3}) satisfying the conclusion above. This method essentially depends on the non-existence result of quasi conformal structure on manifolds with non standard definite intersection form, like |−E_{8}|♯|−E_{8}|. Here one gets the result by constructing directly Yang Mills moduli spaces over M̂.

There may be intrinsic difference between smooth structures of a closed manifold and the complement of one point, M̂ = M\ pt. Recall that by [Q], any topological four manifold admits a smooth structure on any complement of one point. Suppose there are two smooth M and M′, each are homeomorphic. In order to compare smooth structures of the ends between M̂ and M̂′, we will introduce morphisms between M̂ and M̂′ as follows.

Let K_{0}(′) ⊂ K_{1}(′) ⊂ ⋯ ⊂ M̂(M̂′) be two exaustions as before. A (locally bounded) asymptotic morphism from (M̂, g) to (M̂′, g′) consists of a family of quasiconformal homeomorphisms I_{l} : (K_{l}, g) ≅ (K′_{l}, g′) and δ > 0 satisfying:

(1) |H(I_{l})^{2} − 1|L_{loc}^{N} → 0 for a sufficiently large N >> 0,

(2) d(I_{l′}(x), I_{l}(x)) → 0 as l, l′ → ∞ and

(3) |∇(I_{l′} − I_{l})|L_{loc}^{4+δ}(K_{l}) → 0,

(4) for any x ∈ M̂′, there exists l(x) ≥ 0 such that image of I_{l} contains x for all l ≥ l(x), and I_{l}(m) → ∞ as m → ∞.

A uniformly bounded asymptotic morphism consists of an asymptotic morphism with the addtional conditions:

(5) d(x, I_{l}(x)) are uniformly bounded, where we identify both endM̂ and endM̂′ as S^{3} × [0, ∞),

If the family {I_{l}^{−1}}_{l} consists of another asymptotic morphism, then we say that M̂ and M̂′ are asymptotically q.c. equivalent.

In this paper we shall introduce asymptotic Banach spaces. Using them as the underlying spaces, we construct asymptotic ASD moduli spaces As𝔐(M̂, g). It contains the ordinary ASD moduli space 𝔐(M̂). If these donot coincide, then by the construction it implies the existence of some type of bubblings on the end of 𝔐(M̂). The type is determined by the underlying Banach spaces. As𝔐(M̂, g) = As𝔐(M̂, g; a, T) is determined by choosing some families of indices a = {a(l, j)} and T = {T(l, j)}. Then by gathering, one gets the parametrized asymptotic ASD moduli spaces:

P As𝔐(M̂, g) = ∪_{a,T}As𝔐(M̂, g; a, T).

Theorem 0.2 (1) Suppose there exists a locally bounded equivalence as above, such that for any [A_{l}] ∈ As𝔐(M̂(′), g(′); a, T),

lim_{r→0}lim sup_{x→∞} sup_{k,l}|I_{l}(^{−1})^{∗}(F_{A_{k}})|L^{2}(B_{r}(x)) = 0. ∗

Then it induces a homeomorphism:

{I_{l}^{∗}}_{l} : P As𝔐(M̂, g) ≅ P As𝔐(M̂′, g′)

where g and g′ are sufficiently near the cylindrical ones.

(2) There is a natural embedding:

𝔐(M̂, g) ↪ As𝔐(M̂, g; a, T).

Thus the connected components of these spaces may land on different kinds of another by {I_{l}^{∗}}. Notice that we donot assume genericity of Riemannian metrics. This is verified in section 7. As we will see in 2.A, the underlying Banach spaces are different from the ordinary ones. It would be very interesting to use non uniformly bounded Banach spaces (ϵ collapsed one, 3.C). Fredholm property works for the spaces, however gauge group action seems to be problematic.

The author expects that one can construct larger asymptotic ASD moduli spaces Ās𝔐 so that it contains the above asymptotic ASD moduli space As𝔐, and it could contain the information on the local bubbling phenomena by mapping by I_{l}. In particular if one uses Ās𝔐 in theorem 0.2, one would not need to exclude local bubbling situation (∗ above).

When we consider bundles with small p_{1}(E), then the condition ∗ is automatically satisfied. Let {I_{l}}_{l} be a locally bounded asymptotic equivalence between (M̂, g) and (M̂′, g′). Let us take isomorphic SO(3) bundles E and E′ over M̂ and M̂′ with I = −2p_{1}(E) − 3(1 + b_{2}^{+}(M)) > 0.

Theorem 0.3 Suppose 1 ≤ I ≤ 12. Then the induced morphism:

{I_{l}^{∗}}_{l} : P As𝔐(E, M̂, g) ≅ P As𝔐(E′, M̂′, g′)

gives a homeomorphism.

Moreover for −p_{1}(E) ≤ 3, As𝔐(E, M̂, g) is homeomorphic to the compact ASD moduli space 𝔐(E, M̂, g).

Corollary 0.1 If the Donaldson’s invariants of M and M′ are different from each other, then they cannot be asymptotically q.c. equivalent.

These notions all depend on the Riemannian metric g on M̂. However these properties hold under C^{0} perturbations of g in the following sense. Let φ : M̂ ↦ Gl(T M̂) be any bounded continuous section. Then for any Riemannian metrics g and g′, one has the equality d([g], [g′]) = d([φ^{∗}(g)], [φ^{∗}(g′)]). In particular if φ = dΨ for some diffeomorphism Ψ near id, and if there is an ACH from M̌ to (M̂, g), then one gets a perturbed ACH from M̌ to (M̂, φ^{∗}(g)).

Here we point out that asymptoticity of ∗ homomorphisms are elegantly introduced in the C^{∗} algebras world in [CH].

The author would like to express his gratitude to the Max Planck Institut during making essential steps of this work, where he found very warmful atmosphere. He also would be thankful to IHES during his stay there.

1 Function spaces

1.A Function spaces: In this paper we always assume that M is simply connected. Let (M̂, g) be a complete Riemannian four manifold whose end is isometric to (S^{3} × [0, ∞), g|S^{3} × dt^{2}), where g|S^{3} is a fixed smooth metric on S^{3}. Let us take a positive number δ > 0, and take a smooth function w : M̂ ↦ [0, ∞) with w| end (t, m) = tδ. Sometimes we denote w = w(δ) to specify the constant. Recall the weighted Sobolev space L_{w}^{p}(M̂) is a Banach space with norm |u|^{p} = ∫M̂ exp(w)|u|^{p}. Similarly one defines weighted Sobolev k spaces as (W_{k}^{p})_{w}(M̂) = {u : ∇^{i}(u) ∈ L_{w}^{p}(M̂; Λ^{∗}), i ≤ k}.

Let us take a small ϵ ≥ 0 and a pair of positive constants 0 < δ′ ≤ δ. We put w′ = w(δ′) and w = w(δ). Then one defines particular Banach spaces by:

B^{0}(M̂) = {u ∈ L_{2w′}^{4+2ϵ}(M̂), du ∈ L_{2w}^{4+2ϵ}(M̂)},
B^{1}(M̂) = {u ∈ L_{2w}^{4+2ϵ}, d(u) ∈ L_{w}^{2+ϵ}}
B^{2}(M̂) = L_{w}^{2+ϵ}(M̂; Λ_{+}^{2}).

1.B AHS complex over Y × R: In this section one considers the Atiyah Hitchin Singer complex where δ′ = δ and d^{+} = (1 + ∗)d:

0 ⟶ B^{0}(M̂) ⟶^{d} B^{1}(M̂) ⟶^{d^{+}} B^{2}(M̂) ⟶ 0.  (1.1)

Here we have a simple lemma:

Lemma 1.1 If the AHS complex above is of Fredholm, then the same is true for the pair 0 < δ′ < δ.

This follows, since the bounds hold:

|du|B^{1}(M̂) = |du|L_{2w}^{4+2ϵ}(M̂) ≥ C|u|L_{2w}^{4+2ϵ}(M̂) ≥ C|u|L_{2w′}^{4+2ϵ}(M̂).

We will choose a smaller δ′ < δ in 3.D.

Let (Y, g) be a Riemannian 3 manifold and denote also by g as the product one on Y × R. Let Λ^{1}(Y × R) and Λ_{+}^{2}(Y × R) be the exterior differentials of 1 forms and self dual 2 forms with respect to g. Then we have the natural identification:

Λ^{1}(Y × R) = p^{∗}(Λ^{1}(Y)) ⊕ p^{∗}(Λ^{0}(Y)),
Λ_{+}^{2}(Y × R) = p^{∗}(Λ^{1}(Y))

where p : Y × R ↦ Y is the projection. The isomorphisms are given by:

u + vdt ↔ (u, v), ∗_{Y} u + u ∧ dt ↔ u.

Let us put X = Y × R. Using L^{2} adjoint operator, we get the elliptic operator P = d^{∗} ⊕ d^{+} : Λ^{1}(X) ↦ Λ^{0}(X) ⊕ Λ_{+}^{2}(X). Passing through the identification, this is expressed as:

P = d^{∗} ⊕ d^{+} : p^{∗}(Λ^{0}(Y) ⊕ Λ^{1}(Y)) ↦ p^{∗}(Λ^{0}(Y) ⊕ Λ^{1}(Y)).

Let us use t as the coordinate on R. Then one has the following expression:

P = −\frac{d}{dt} + (\begin{matrix}∗_{Y} d & d \\ d^{∗} & 0 \end{matrix}) ≡ −\frac{d}{dt} + Q.

Notice that Q is an elliptic self adjoint differential operator on L^{2}(Y; Λ^{0}(Y) ⊕ Λ^{1}(Y)). Let us define an isometry (m = 2, 4):

I : L^{m+ϵ}(M̂, Λ^{∗}) ↦ L_{w}^{m+ϵ}(M̂, Λ^{∗})

by I(u) = exp(−\frac{w}{m+ϵ})u. Then a simple calculation shows the equality:

I^{−1}P I = P + \frac{1}{m + ϵ} \frac{dw}{dt} : W_{k+1}^{m+ϵ}(M̂; Λ^{1})) ↦ W_{k}^{m+ϵ}(M̂; Λ^{0} ⊕ Λ_{+}^{2})

P (w) ≡ I^{−1}P I = −\frac{d}{dt} + (\begin{matrix}∗_{M}d & d \\ d^{∗} & 0 \end{matrix}) + \frac{1}{m + ϵ} \frac{dw}{dt} ≡ −\frac{d}{dt} + Q_{w} on the end.

One has a similar expression for P^{∗}. Let us put the operator:

Q(δ) = (\begin{matrix}∗_{M}d − \frac{δ}{m+ϵ} & d \\ d^{∗} & −\frac{δ}{m+ϵ} \end{matrix}) : W_{k+1}^{m+ϵ}(Y, Λ^{1} ⊕ Λ^{0}) ↦ W_{k}^{m+ϵ}(Y, Λ^{1} ⊕ Λ^{0}).

Notice that Q(−δ) coincides with Q_{w} on the end. We put Q(δ) = P (−δ). There is a similar expression for P^{∗} = d ⊕ (d^{+})^{∗}. In particular P^{∗}(δ) = Q(δ).

(d^{+})^{∗} is the formal adjoint operator with respect to the L^{2} inner product. We will denote by (d^{+})_{w}^{∗} as the one for L_{w}^{2} inner product.

1.C Computation of cohomology groups: This section is occupied to verify the following:

Proposition 1.1 For a small choice of δ > 0 and 0 ≤ ϵ ≤ ϵ_{0}, the AHS complex is of Fredholm with H^{0} = H^{1} = 0 and dimH^{2} = b_{+}^{2}(M).

P roof : First let us consider i = 0. We show that d : B^{0}(M̂) ↦ B^{1}(M̂) has closed range. Recall Q(δ) is an elliptic operator on L^{m+ϵ}(S^{3}) defined using g|S^{3}. We show that:

P^{∗} = d ⊕ (d^{+})^{∗} : (W_{1}^{4+ϵ})_{2w}(M̂; Λ^{0} ⊕ Λ_{+}^{2}) ↦ L_{2w}^{4+ϵ}(M̂; Λ^{1})

has closed range with finite dimensional kernel.

Sublemma 1.1 (1) Let us choose a small δ > 0 and any 0 ≤ ϵ ≤ ϵ_{0}. We put p = 4 + 2ϵ or 2 + ϵ. Then:

(1) P^{∗}(δ) = Q(δ) : W_{1}^{p}(S^{3}) ↦ L^{p}(S^{3}) satisfies |Q(δ)| ≥ C > 0 for some C = C(δ).

(2) Q(δ) + \frac{d}{dt} : W_{1}^{p}(S^{3} × R) ↦ L^{p}(S^{3} × R) is invertible ([LM]).

P roof : (1) Let us take a family u_{i} ∈ W_{1}^{4+2ϵ}(S^{3}) satisfying |u_{i}| = 1 and |Q(δ)u_{i}|L^{4+2ϵ}(S^{3}) → 0, i → ∞. ¿From the three dimensional Sobolev embedding, the inclusion W_{1}^{4+2ϵ}(S^{3}) ↪ L^{4+2ϵ}(S^{3}) is compact (see [GT] p167), and a subsequence of {u_{i}}_{i} converges to u_{∞} ∈ L^{4+2ϵ}(S^{3}). On the other hand, by the elliptic estimate, there is a positive constant C > 0 with |u_{i}|L^{4+2ϵ}(S^{3}) > C. In particular u_{∞} is non zero. Again by the elliptic estimate, u_{∞} lies in W_{1}^{4+2ϵ}(S^{3}) and satisfies Q(δ)u_{∞} = 0. By elliptic regularity, u_{∞} is smooth, but there are no such solution over S^{3}.

For p = 2 + ϵ case, the process is similar. This shows the result.

Remark 1.1: In proposition 3.2, we will verify a similar result as (2) above, using different kinds of Banach spaces.

Let us consider (2). Let us put Δ = Q(δ)^{2}, A = −\frac{d}{dt} + Q(δ) and A^{∗} = \frac{d}{dt} + Q(δ). Then we consider:

A^{∗}A = AA^{∗} = −\frac{d^{2}}{dt^{2}} + Δ : W_{2}^{4+2ϵ}(S^{3} × R) ↦ L^{4+2ϵ}(S^{3} × R).

It follows from the estimate below that in order to verify (2), It is enough to see that the above operator is invertible:

|A^{∗}(u)|L^{4+2ϵ} = |A^{∗}A(v)|L^{4+2ϵ} ≥ C|v|W_{2}^{4+2ϵ} ≥ C|u|W_{1}^{4+2ϵ}

where v = A^{∗}(AA^{∗})^{−1}(u) ∈ W_{2}^{4+2ϵ}.

Proof of proposition: Let us choose a cut off function φ : M̂ ↦ [0, 1] with Suppdφ ⊂ [0, 1] × S^{3} ⊂ M̂. Then one has the estimate:

|u|(W_{1}^{4+ϵ})_{2w} ≤ |φu|(W_{1}^{4+ϵ})_{2w} + |(1 − φ)u|(W_{1}^{4+ϵ})_{2w}
≤ |φu|(W_{1}^{4+ϵ})_{2w} + C|P^{∗}[(1 − φ)u]|L_{2w}^{4+ϵ}

≤ C{|φu|(W_{1}^{4+ϵ})_{2w} + |(1 − φ)P^{∗}(u)|L_{2w}^{4+ϵ} + |[1 − φ, P^{∗}]u|L_{2w}^{4+ϵ}},

|φu|(W_{1}^{4+ϵ})_{2w} ≤ C{|φP^{∗}(u)|L_{2w}^{4+ϵ} + |φu|L_{2w}^{4+ϵ} + |[φ, P^{∗}]u|L_{2w}^{4+ϵ}.

Notice Suppφ, Supp[1 − φ, P^{∗}] and Supp[φ, P^{∗}] are all compact.

Since the inclusion W_{1}^{4+2ϵ}(K) ↪ L^{4+2ϵ}(K) is compact for all ϵ ≥ 0 and for any compact subset K, it follows that P^{∗} has closed range with finite dimensional kernel. In particular d : B^{0} ↦ B^{1} has closed range, since |df|B^{1} = |df|L_{2w}^{4+ϵ}. Thus one has verified the result H^{0} = 0 for ϵ ≥ 0.

Next we show H^{1} = 0.

Sublemma 1.2 (T) Let f satisfy df ∈ L_{w}^{p}. Then there is a constant c uniquely determined by f satisfying the following estimates:

[∫ exp(\frac{4w}{4 − p})|f − c|^{\frac{4p}{4−p}}]^{\frac{4−p}{4}} ≤ C(p, δ) ∫ exp(w)|df|^{p}, p ∈ (2, 4),

(∫ exp(2w)|f − c|^{4})^{\frac{1}{2}} ≤ C ∫ exp(w)|df|^{2},

|f − c|C^{0}(S^{3} × \{t\}) ≤ C(t)exp(−\frac{δ}{4 + ϵ}t), p = 4 + ϵ > 4, C(t) → 0.

Let us take u ∈ H^{1}. Thus d^{+}(u) = 0 holds. Notice that for any positive μ > 0, exp(−μw) is integrable on M̂. Then using Holder’s inequality, one sees d(u) ∈ L^{2}(M̂; Λ^{2}). d(u) = 0 holds by integrating by parts. Let us consider the complex {(W_{k}^{p})_{loc}(M̂; Λ^{∗}), d_{∗}} and their cohomology groups H^{∗}. In general in order to get the isomorphism of H^{∗} with the usual de Rham cohomology groups, it is enough to check existence of:

(1) partition of unity in (W_{k}^{p})_{loc} spaces and

(2) Poincaré lemma in (W_{k}^{p})_{loc}(D; Λ^{∗}) for small disks D (see [W]).

Thus there is f ∈ (W_{1}^{4+2ϵ})_{loc}(M̂) with df = u. Then by the above sublemma, u lies in the closure of d((W_{1}^{4+2ϵ})_{2w}(M̂)) ⊂ B^{1}(M̂). Since d has closed range, this implies that there is f ∈ L_{2w}^{4+2ϵ} with df = u. This shows H^{1} = 0.

Let us consider the case H^{2}. Sublemma 1.3 For all 0 ≤ δ ≤ δ_{0} and all 0 ≤ ϵ ≤ ϵ_{0}, d^{+} : (W_{1}^{2+ϵ})_{w}(M̂; Λ^{1}) ↦ L_{w}^{2+ϵ}(M̂; Λ_{+}^{2}) has closed range with finite dimensional cokernel. P roof : Recall that self dual 2 forms over S^{3} × R can be regarded as 1 forms over S^{3}. Let (d^{+})^{∗} be the formal adjoint operator (without weight function). Then one has the equality:

Δ = d^{+} ∘ (d^{+})^{∗} = −\frac{d^{2}}{dt^{2}} + Δ_{1}^{3} : W_{2}^{2+ϵ}(S^{3} × R; p^{∗}Λ^{1}) ↦ L^{2+ϵ}(S^{3} × R; p^{∗}Λ^{1})

where Δ_{1}^{3} is the Laplace operator on 1 forms over S^{3}. This operator has closed range with null kernel since Δ_{1}^{3} is invertible over S^{3} (see also proposition 3.2).

It is also true for Δ_{1}^{3} ± δ for all small δ. In particular the same argument as function case shows that d^{+} ∘(d^{+})_{w}^{∗} : (W_{2}^{p})_{w}(M̂; Λ_{+}^{2}) ↦ L_{w}^{p}(M̂; Λ_{+}^{2}) also satisfies closed range and finite dimensional kernel. (here one uses compactness of the inclusion W_{2}^{2+ϵ} ↪ W_{1}^{2+ϵ}).

Let us see d^{+} has finite dimensional cokernel. At ϵ = 0, cokerneld^{+} = ker(d^{+})_{w}^{∗} holds using Hilbert space structure. We show the same property for all ϵ. Any element u ∈ H= Ker(d^{+})_{w}^{∗} ∩ L_{w}^{2} lies in also L_{w}^{2+ϵ}, since it satisfies exponential decay estimate (see sublemma 3.1 or the inequality below).

Let us take u ∈ CokerΔ ∩ L_{w}^{2+ϵ}. We choose a family of cut off functions φ_{i} with exausting supports. Then each u_{i} = φ_{i}u lies also in L_{w}^{2}. Thus there are v_{i} ∈ H and α_{i} ∈ Kerd_{w}^{∗} ∩ (W_{1}^{2})_{w} with u_{i} = d^{+}(α_{i}) + v_{i} ∈ L_{w}^{2} ∩ L_{w}^{2+ϵ}. Now we claim α_{i} ∈ (W_{1}^{2+ϵ})_{w}. For this it is enough to see α_{i} ∈ L_{w}^{2+ϵ} by the elliptic estimate. Since Δ has closed range, this implies u ∈ H.

Let us verify the claim. The following argument will be called later as a local to global method. By normalizing one makes |α_{i}|(W_{1}^{2})_{w} = 1. Recall endM̂ is isometric to S^{3} × [0, ∞). By the Sobolev inequality, there is a constant C with:

C^{−1}|α_{i}|L^{2+ϵ}(S^{3} × [n, n + 1]) ≤ |α_{i}|W_{1}^{2}(S^{3} × [n, n + 1]) ≤ exp(−\frac{nδ}{2}) ≤ 1.

Then we have inequality:

|α_{i}|^{2+ϵ}L_{w}^{2+ϵ} = Σ_{n}exp(nδ)|α_{i}|^{2+ϵ}L^{2+ϵ}(S^{3} × [n, n + 1])
≤ CΣ_{n}exp(nδ)|α_{i}|^{2+ϵ}W_{1}^{2}(S^{3} × [n, n + 1])
≤ CΣ_{n}exp(nδ)|α_{i}|^{2}W_{1}^{2}(S^{3} × [n, n + 1])
= C|α_{i}|^{2}(W_{1}^{2})_{w} = C|α_{i}|^{2+ϵ}(W_{1}^{2})_{w}.

Thus we get an inequality |α_{i}|L_{w}^{2+ϵ} ≤ C|α_{i}|(W_{1}^{2})_{w}. Again by multiplying constants, one may remove the norm condition above. Thus this inequality holds in general. This verifies the claim.

Sublemma 1.4 Let Q′ : L_{w}^{2}(M̂)∩ imd^{+} ↦ (W_{1}^{2})_{w}(M̂) be the inverse to d^{+}. Then there is an extension Q : L_{w}^{2+ϵ}∩ imd^{+} ↦ (W_{1}^{2+ϵ})_{w}.

P roof : Recall that d^{+}((W_{1}^{2+ϵ})_{w}) ⊂ L_{w}^{2+ϵ} has finite codimension for all sufficiently small 0 ≤ δ ≤ δ_{0} and small 0 ≤ ϵ. Let us denote H(δ, ϵ) = Cokerd^{+} ⊂ L_{w}^{2+ϵ}(M̂; Λ_{+}^{2}). Any element u ∈ H(0) = H(0, 0) lies in (W_{k}^{2+ϵ})_{w} for all k. This follows since u satisfies the exponential decay estimate (see sublemma 3.1) and Sobolev estimate. First we claim H(0) ⊂ H(δ, ϵ). Suppose not. Then there is u ∈ H(0) satisfying u = d^{+}(α), α ∈ (W_{1}^{2+ϵ})_{w}. Let us take another δ′ < δ and the corresponding weight function w′. We see α ∈ (W_{1}^{2})_{w′} below. This shows u = 0 since H(δ, 0) is unchanged under small perturbations of 0 ≤ δ ≤ δ_{0}. Recall the Holder’s inequality ∫ ab ≤ (∫ a^{p})^{\frac{1}{p}}(∫ b^{q})^{\frac{1}{q}} for \frac{1}{p} + \frac{1}{q} = 1. Then we apply this for

ab = exp(w′)|α|^{2}, b^{q} = exp(w)|α|^{2+ϵ}.

Then we solve a = exp(w′ − q^{−1}w)|α|^{2−q^{−1}(2+ϵ)}. Letting q = \frac{(2+ϵ)}{2}, one has a = exp(w′ − q^{−1}w). With respect to ϵ, one chooses a sufficiently small δ′ < δ so that δ′ − q^{−1}δ < 0. Then a^{p} is integrable. This shows exp(w′)|α|^{2} is also integrable. Thus we get the claim H(0) ⊂ H(δ, ϵ).

Next we show the converse direction. Let us consider ∇ = d_{w′}^{∗} ⊕ d^{+} : (W_{1}^{2+ϵ})_{w′} ↦ L_{w′}^{2+ϵ}. We claim that for δ′ > 0, there is a constant C > 0 with bound:

|u|(W_{1}^{2+ϵ})_{w′} ≤ C|∇(u)|L_{w′}^{2+ϵ}.

Notice that Δ_{w} = d_{w}^{∗}d : (W_{2}^{2+ϵ})_{w} ↦ L_{w}^{2+ϵ} gives an isomorphism on functions. In particular d_{w′}^{∗} : (W_{1}^{2+ϵ})_{w′} ↦ L_{w′}^{2+ϵ} has closed range. Since d^{+} also has closed range, it is enough to check Ker∇ = 0. Let us take α ∈ Ker∇. On the endM̂, I^{−1}∇I : W_{1}^{2+ϵ} ↦ L^{2+ϵ} has the following form where I is in 1.B:

−\frac{d}{dt} + (\begin{matrix}∗_{M}d + \frac{δ′}{m+ϵ} & d \\ d^{∗} & −δ′ \frac{m+ϵ−1}{m+ϵ} \end{matrix}) ≡ −\frac{d}{dt} + Q′(δ′).

Notice that Q′(δ′) is a self adjoint elliptic operator. Moreover for small δ′, there are no kernels (see [K] lemma 6.3). Thus I^{−1}(α) ∈ W_{1}^{2+ϵ} satisfies the exponential decay (see sublemma 3.1). In particular one sees α ∈ (W_{1}^{2+ϵ})_{w}.

Now as before one has d(α) = 0. Let us put m = \frac{4}{2−ϵ}. Then by sublemma 1.2, there exists f ∈ L_{mw}^{m(2+ϵ)} with df = α. One may assume f ∈ L_{w′}^{2}. Since f satisfies d_{w′}^{∗} ∘ df = 0, this implies df = α = 0. This shows the claim.

Suppose H(0) ≠ H(δ, ϵ). Thus there is u ∈ H(δ, ϵ) but u ≠ H(0). As above, one may assume u ∈ L_{w′}^{2}. Then one may write u = d^{+}(α), α ∈ (W_{1}^{2})_{w′} and d_{w′}^{∗}(α) = 0. ¿From the Sobolev embedding (W_{1}^{2})_{loc} ↪ L_{loc}^{2+ϵ}, u_{i} = d^{+}(φ_{i}α) and μ_{i} = d_{w′}^{∗}(φ_{i}α) converges to u and zero respectively in L_{w′}^{2+ϵ}. ¿From the claim, one sees α ∈ (W_{1}^{2+ϵ})_{w′}. This shows that the image of H(δ, ϵ) ↦ H(δ′, ϵ) is exactly H(0).

We claim that for all sufficiently small pairs (δ, ϵ), the above maps are all isomorphisms. Suppose not. Then for all small δ, one has non zero elements in H(δ, ϵ)\H(0). Let Q(δ, ϵ) : L_{w}^{2+ϵ} ↦ (W_{1}^{2+ϵ})_{w} satisfy d^{+} ∘ Q(δ, ϵ)| imd^{+} is the identity and Q(δ, ϵ)|H(δ, ϵ) = 0. We denote Q = Q(0, ϵ). Passing through the isomorphisms I(δ) : (W_{k}^{2+ϵ})_{w} ≅ W_{k}^{2+ϵ}, one may assume that all the operators act on W_{k}^{2+ϵ}. Let us denote d^{+}(δ) = I(δ)d^{+}I(δ)^{−1}. Notice that d^{+}(δ) is a continuous operators on W_{1}^{2+ϵ} with respect to δ. Then the following holds:

|d^{+}(δ)Q − id|L^{2+ϵ}\H(0) → 0 δ → 0.

In particular for any sequence δ_{i}, δ_{i} → 0, there are vectors u_{i} ∈ H(δ_{i}, ϵ)\H(0), |u_{i}|L^{2+ϵ} = 1 which satisfy |d^{+}(δ_{i})Q(u_{i}) − u_{i}|L^{2+ϵ} → 0 (∗).

Next let us consider a function:

δ′ ∈ [0, δ] ↦ d( im d^{+}(δ′), Coker ) = Sup_{|u|L^{2+ϵ}=1} inf_{f}|u − d^{+}(δ′)f|L^{2+ϵ}.

Clearly this is continuous. In particular there is a positive constant C independent of δ′ with:

inf_{f}|u_{i} − d^{+}(δ′)f|L^{2+ϵ} > C.

This contradicts with (∗). This shows that for sufficiently small ϵ ∈ [0, ϵ_{0}] and 0 ≤ δ ≤ δ_{0}, H(ϵ, δ) = H(0, 0). Notice that one knows d^{+}((W_{1}^{2})_{w}) ⊂ L_{w}^{2} has codimension b_{+}^{2}(M) ( [T]). In particular d^{+}((W_{1}^{2+ϵ})_{w}) ⊂ L_{w}^{2+ϵ} has codimension b_{+}^{2}(M).

Recall that Δ_{w} = d_{w}^{∗}d : (W_{2}^{2+ϵ})_{w} ↦ L_{w}^{2+ϵ} gives an isomorphism. Let us take α ∈ Kerd^{+} ⊂ (W_{1}^{2+ϵ})_{w}(M̂; Λ^{1}). Then d(d_{w}^{∗}d)^{−1}d_{w}^{∗}(α) = α since |∇| ≥ C > 0. In particular Ker d^{+} is equal to imd and H^{1} = 0. Then by the open mapping theorem, the following map gives an isomorphism between Banach spaces:

dΔ_{w}^{−1}d_{w}^{∗} ⊕ d^{+} : (W_{1}^{2+ϵ})_{w} ↦ Kerd^{+} ⊂ (W_{1}^{2+ϵ})_{w} ⊕ imd^{+} ⊂ L_{w}^{2+ϵ}.

(Continuity and injectivity are clear. For surjectivity, let us take (u = df, d^{+}(v)) ∈ Kerd^{+}⊕ imd. Then u − dΔ_{w}^{−1}d_{w}^{∗}(v) + v maps to (u, d^{+}v)).)

Let H ⊂ L_{w}^{2+ϵ} be the finite dimensional subspace which represents the cokernel of d^{+}. Then since H is finite dimensional, one has a natural isomorphism between Banach spaces L_{w}^{2+ϵ} ≅ imd^{+} ⊕ H. Let us define a map Q by Q(d^{+}(v)) = v − dΔ_{w}^{−1}d_{w}^{∗}(v) for v ∈ (W_{1}^{2+ϵ})_{w}. Then this gives a bounded map:

Q : L_{w}^{2+ϵ}(M̂; Λ_{+}^{2}) ∩ imd^{+} ↦ Kerd_{w}^{∗} ∩ (W_{1}^{2+ϵ})_{w}(M̂; Λ^{1})

which satisfies d^{+}Q| imd^{+} = id. This completes the proof of sublemma 1.4.

Proof of proposition: Using the Sobolev inequality and the above Q, one has the estimate:

|u|^{4+2ϵ}L_{w}^{2+ϵ} ≥ C|Q(u)|^{4+2ϵ}(W_{1}^{2+ϵ})_{w}

= \frac{C}{2} |Q(u)|^{4+2ϵ}(W_{1}^{2+ϵ})_{w} + \frac{C}{2} |Q(u)|^{4+2ϵ}(W_{1}^{2+ϵ})_{w}

≥ C′|u|^{4+2ϵ}L_{w}^{2+ϵ} + C′(Σ_{i∈N}exp(iδ)|Q(u)|^{2+ϵ}W_{1}^{2+ϵ}([i, i + 1] × S^{3}))^{2} ≥ C′|u|^{4+2ϵ}L_{w}^{2+ϵ} + C′(Σ_{i∈N}exp(iδ)|Q(u)|^{2+ϵ}L^{4+2ϵ}([i, i + 1] × S^{3}))^{2} ≥ C′|Q(u)|^{4+2ϵ}B^{1}.

This shows that one has a continuous extension:

Q : H^{⊥} ⊂ L_{w}^{2+ϵ}(M̂; Λ_{+}^{2}) ↦ B^{1}

satisfying d^{+}Q is the identity. In particular this shows that d^{+} : B^{1} ↦ L_{w}^{2+ϵ} has closed range.

Recall Cokerd^{+} : (W_{1}^{2+ϵ})_{w} ↦ L_{w}^{2+ϵ} has b_{+}^{2}(M) dimension. Since Q : L_{w}^{2+ϵ} ↦ (W_{1}^{2+ϵ})_{w} has an extension to Q : L_{w}^{2+ϵ} ↦ B^{1} with Q|H = 0 and d^{+}Q| imd^{+} is the identity, it follows dimH^{2} ≤ b_{+}^{2}(M). Suppose there is a vector α ∈ L_{2w}^{4+2ϵ} with d^{+}(α) ∈ H. Then as before α lies in L_{w′}^{2}. One may assume inf_{f}|α − df|L_{w′}^{2} = |α|L_{w′}^{2}. ¿From this d_{w′}^{∗}α = 0 (differentiate |α + tdf|^{2}L_{w′}^{2}). Then from the elliptic estimate, it follows α ∈ (W_{1}^{2})_{w′}. This shows α = 0. This completes the proof.

Remark 1.2: One gets a relation of the range of Q and an inequality:

\begin{matrix}imQ ⊂ B^{1}(M̂, ϵ) ∩ (W_{1}^{2+ϵ})_{w}(M̂) ∩ Kerd_{w}^{∗}, \\ |d^{−}Qu|L_{w}^{2+ϵ}(M̂) ≤ C|d^{+}Qu|L_{w}^{2+ϵ}(M̂). \end{matrix}

2 Asymptotic function spaces

2.A Asymptotic Banach spaces: Here we introduce asymptotic function spaces. First we define the asymptotic Banach spaces B̂^{∗}(M̂; 1) with one bubble. Then we generalize them to another B̂^{∗}(M̂; n) with n bubbles. Later when we construct moduli spaces, we will use B̂^{∗}(M̂) ≡ B̂^{∗}(M̂; n) for some n > 0 as the underlying Banach spaces.

Let (M̂, g) be a cylindrical manifold as before, whose end is isometric to (S^{3} × [0, ∞), g|S^{3} + dt^{2}). Let us take an exaustion by compact subsets:

K_{0} ⊂ K_{1} ⊂ ⋯ ⊂ M̂.

Here one always assume that the exaustion is of bounded geometry (boundaries are uniformly smooth). For each l ≥ 0, let us take a cut off function φ_{l} : M̂ ↦ [0, 1] satisfying:

(1) φ_{l}|M̂\S^{3} × [l, ∞) ≡ 1, (2) φ_{l}|S^{3} × [l + 1, ∞) ≡ 0,
(3) |φ_{l}|C^{i} are bounded independently of l, i ≥ 0.

By sliding indices, if necessarily, one may assume ∂K_{l} ⊂ S^{3}×[3l, ∞) ⊂ end M̂.

Let K ⊂ M̂ be a compact subset with smooth boundary ∂K and intK be its interior. Then here one uses a notation (W_{k}^{p})_{w}(K)_{0} to be a Banach space where C_{cp}^{∞}( intK) is dense in (W_{k}^{p})_{w}(K)_{0} with norm |u|^{p}(W_{k}^{p})_{w}(K)_{0} = Σ_{i≤k} ∫_{K} exp(w)|∇^{i}u|^{p}. Notice the inclusion (W_{k}^{p})_{w}(K)_{0} ⊂ (W_{k}^{p})_{w}(M̂). One defines similarly B^{i}(K)_{0} using L_{w}^{p}(K)_{0} as before. These also have inclusions B^{i}(K)_{0} ⊂ B^{i}(M̂).

Let us fix small constants ϵ > 0, δ ≥ δ′ > 0 and the corresponding weight functions w and w′. Then we introduce thickened Banach spaces:

|u|L̂_{2w}^{4}(M̂) ≡ Sup_{0≤ϵ′≤ϵ}|u|L_{2w}^{4+2ϵ′}(M̂).

Similarly we define L̂_{w}^{2} = Sup_{0≤ϵ′≤ϵ}| |L_{w}^{2+ϵ′}(M̂), or (Ŵ_{k}^{2})_{w}. Then we put:

\begin{matrix}B̄^{0}(M̂, w, ϵ) = B̄^{0}(M̂) = \{u ∈ L̂_{2w′}^{4}(M̂), du ∈ L̂_{2w}^{4}(M̂)\}, \\ B̄^{1}(M̂) = \{u ∈ L̂_{2w}^{4}(M̂), d(u) ∈ L̂_{w}^{2}(M̂)\}, B̄^{2}(M̂) = L̂_{w}^{2}(M̂; Λ_{+}^{2}). \end{matrix}

Let w_{l} : S^{3} × R ↦ [0, ∞) be a weight function defined by w_{l}(m, t) = δ|l − t|. We will use the same notation for the restriction w_{l} : S^{3} × R_{+} = endM̂ ↦ [0, ∞). Moreover let T_{l} : S^{3} × [−l, ∞)(⊂ S^{3} × R) ↦ S^{3} × [0, ∞) ⊂ ( end M̂) be the linear translation defined as T_{l}(m, k) = (m, k + l).

Let us consider the set [u_{l}], u_{l} ∈ B̄^{∗}(K_{l})_{0} satisfying the following:

(1) φ_{l}u_{l} converges to u_{1} ∈ B̄^{∗}(M̂, w, ϵ),

(2) (1 − φ_{l})u_{l} ∈ B̄(M̂, w_{2l}, ϵ) and T_{2l}^{∗}((1 − φ_{l})u_{l}) converges to u_{2} ∈ B̄^{∗}(S^{3} × R, w_{0}, ϵ).

Then one defines linear spaces B̄^{i}({K_{l}}_{l}) as follows:

B̄^{i}({K_{l}}_{l}) = {[u_{l}] : satisfying the above (1), (2)}.

Let us equip a norm on B̄^{i}({K_{l}}_{l}) as:

|[u_{l}]|B̄^{∗}({K_{l}}_{l}) = Sup_{l}||u_{l}||_{l}
≡ Sup_{l}{|φ_{l}u_{l}|B̄^{∗}(M̂, w, ϵ) + |(1 − φ_{l})u_{l}|B̄^{∗}(M̂, w_{2l}, ϵ)}

= Sup_{l}{|φ_{l}u_{l}|B̄^{∗}(M̂, w, ϵ) + |T_{2l}^{∗}(1 − φ_{l})u_{l}|B̄^{∗}(S^{3} × R, w_{0}, ϵ)} Sometimes we use the notation:

\begin{matrix}||u_{l} − u_{l′}||B̄^{∗} ≡ & |φ_{l}u_{l} − φ_{l′}u_{l′}|B̄^{∗}(M̂, w, ϵ) \\ + |T_{2l}^{∗}(1 − φ_{l})u_{l} − T_{2l′}^{∗}(1 − φ_{l′})u_{l′}|B̄^{∗}(S^{3} × R, w_{0}, ϵ). \end{matrix}

Let us define a vector subspace B̄_{0}^{i}({K_{l}}_{l}) ⊂ B̄^{i}({K_{l}}_{l}):

B̄_{0}^{i}({K_{l}}_{l}) = {[u_{l}] ∈ B̄^{i}({K_{l}}_{l}) : ||u_{l}||_{l}B̄^{i}(K_{l})_{0} → 0 as l → ∞}.

Lemma 2.1 With respect to the above metric, (1) B̄^{i}({K_{l}}_{l}) is a Banach space, (but is not reflexive) and

(2) B̄_{0}^{i}({K_{l}}_{l}) is a closed subspace of B̄^{i}({K_{l}}_{l}).

P roof : (1) We check completeness. Suppose a family [u_{l}^{α}] ∈ B̄^{∗}({K_{l}}_{l}) consists of a Cauchy sequence. Thus for every ϵ > 0, there is a large α_{0} with a uniform bound ||u_{l}^{α} − u_{l}^{β}||_{l}B̄^{∗} < ϵ for all α, β ≥ α_{0} and l. For each l, let β → ∞. Then we get some sequence [u_{l}^{∞}]. We claim this gives an element in B̄^{∗}({K_{l}}_{l}). Let us check convergence property. For any small ϵ > 0, one finds α_{0} with the above property. For [u_{l}^{α_{0}}], let us take a large l_{0} so that ||u_{l}^{α_{0}} − u_{l′}^{α_{0}}||B̄^{∗} ≤ ϵ for all l, l′ ≥ l_{0}. Then for l, l′ ≥ l_{0}, one gets the estimate:

||u_{l}^{∞} − u_{l′}^{∞}||B̄^{∗} ≤ ||u_{l}^{∞} − u_{l}^{α_{0}}||B̄^{∗} + ||u_{l}^{α_{0}} − u_{l′}^{α_{0}}||B̄^{∗} + ||u_{l′}^{α_{0}} − u_{l′}^{∞}||B̄^{∗} ≤ 3ϵ.

This shows that [u_{l}^{∞}] satisfies the defining condition above.

(2) Let us take [u_{l}^{α}] ∈ B̄_{0}^{i}({K_{l}}_{l}), α ≥ 0, and suppose sup_{l}||u_{l}^{α}−u_{l}^{β}||_{l}B̄^{i}(K_{l})_{0} → 0 as α, β → ∞. Then for any ϵ > 0, there is α_{0} so that for all α, β ≥ α_{0}, one has the estimate Sup_{l}||u_{l}^{α} − u_{l}^{β}||_{l}B̄^{i}(K_{l})_{0} ≤ ϵ. Thus letting β → ∞, one has:

Sup_{l}||u_{l}^{α} − u_{l}^{∞}||_{l}B̄^{i}(K_{l})_{0} ≤ ϵ.

This shows [u_{l}^{∞}] ∈ B̄_{0}^{i}({K_{l}}_{l}). This verifies lemma.

Let us put:

B̂^{i}({K_{l}}_{l}) ≡ B̄^{i}({K_{l}}_{l})/B̄_{0}^{i}({K_{l}}_{l}).

One equips the quotient norm on B̂^{i}({K_{l}}_{l}). Now we study some basic properties about B̂^{∗}({K_{l}}_{l}).

Let us introduce another norm on B̂^{i}({K_{l}}_{l}):

|[u_{l}]|_{1} = lim sup_{l→∞}||u_{l}||_{l}B̄^{i}(K_{l})_{0}.

We denote by | |_{0} the quotient norm before:

|[u_{l}]|_{0} = inf { sup_{l}||u_{l} + v_{l}||_{l}B̄^{i}(K_{l})_{0} : [v_{l}] ∈ B̄_{0}^{i}({K_{l}}_{l})}.

Lemma 2.2 | |_{0} and | |_{1} are isometric.

P roof : Let us take [u_{l}] ∈ B̂^{i}({K_{l}}_{l}) with |[u_{l}]|_{1} = 1. Then for any ϵ > 0, there is l_{0} with 1 − ϵ ≤ sup_{l≥l_{0}}||u_{l}||_{l}B̄^{i}(K_{l})_{0} ≤ 1 + ϵ. Let us put:

v_{l} = \begin{cases}−u_{l} & l ≤ l_{0} \\ 0 & l > l_{0}. \end{cases}  (2.1)

Then [v_{l}] ∈ B̄_{0}^{i}({K_{l}}_{l}), and one has:

sup_{l}||u_{l} + v_{l}||_{l}B̄^{i}(K_{l})_{0} = sup_{l≥l_{0}}||u_{l}||_{l}B̄^{i}(K_{l})_{0} ≤ 1 + ϵ.

This shows |[u_{l}]|_{0} ≤ 1. Suppose |[u_{l}]|_{0} = 1 − δ, δ > 0. Then there is [v_{l}] ∈ B̄_{0}^{i}(\{K_{l}\}_{l}) with sup_{l}||u_{l} + v_{l}||_{l}B̄^{i}(K_{l})_{0} ≤ 1 − \frac{δ}{2}. Let us take 4ϵ < δ and a large l with:

||v_{l}||_{l}B̄^{i}(K_{l})_{0} ≤ \frac{δ}{4}, ||u_{l}||_{l}B̄^{i}(K_{l})_{0} ≥ 1 − ϵ.

Then one has the estimates:

\begin{aligned}||u_{l}||_{l}B̄^{i}(K_{l})_{0} & ≤ ||u_{l} + v_{l}||_{l}B̄^{i}(K_{l})_{0} + ||v_{l}||_{l}B̄^{i}(K_{l})_{0} \\ & ≤ 1 − \frac{δ}{2} + \frac{δ}{4} = 1 − \frac{δ}{4} < 1 − ϵ. \end{aligned}

This is a contradiction. This shows the claim.

Let us compare B̂^{∗}({K_{l}}_{l}, w, ϵ) and B̄^{∗}(M̂). These are different Banach spaces. In fact one has the following:

Proposition 2.1 There exists a natural isometric embedding:

i : B̄^{∗}(M̂, w, ϵ) ↪ B̂^{∗}({K_{l}}_{l}, w, ϵ)

with closed image, but i is not surjective.

P roof : Let us take u ∈ B̄^{∗}(M̂) and φ_{l} be the family of cut off functions in the first paragraph in 2.A. Then one puts i(u) = [u_{l} = φ_{l}u]. [u_{l}] defines a non zero element in B̂^{∗}({K_{l}}_{l}, w, ϵ). Closedness is clear, since i is isometric by the above lemma (use | |_{1} norm, also see remark 3.1).

Now let us consider a family u_{l} ∈ B̄^{∗}(K_{l})_{0} defined as follows; recall M̂ contains an isometric subspace (S^{3} × [0, ∞), g|S^{3} × dt^{2}). Let us take any u ∈ C_{cp}^{∞}(S^{3} × (0, 1)), and S_{t} : S^{3} × [t, t + 1] ≅ S^{3} × [0, 1] be the isometry by the translation. Then one puts u_{l} = S_{2l}^{∗}(u) ∈ C_{cp}^{∞}(S^{3} × (2l, 2l + 1)). Then it is clear [u_{l}] defines a non zero element in B̂^{∗}({K_{l}}_{l}, w, ϵ), but it is not contained in the image of i. This completes the proof.

It is easy to get the following:

Corollary 2.1 There is an isomorphism between Banach spaces:

B̂^{∗}({K_{l}}_{l}) ≅ B̄^{∗}(M̂, w, ϵ) ⊕ B̄^{∗}(S^{3} × R, w, ϵ).

B̂^{∗}({K_{l}}_{l}) defined above, is an extended Banach space containing with one bubble. Let us extend them so that they contain n bubbles, n ≥ 1. We change indices of {K_{l}}_{l} by {K_{m(l)}}_{l}, m(l) >> l, if necessarily. Let us take families of indices {a(i, j)}_{i,j} and {b(i, j)}_{i,j}, i = 0, 1, …, j = 0, 1, …, n satisfying:

(1) b(i, j) < a(i, j) < b(i, j + 1) for each j, a(i, n) = ∞,

(2) a(i, j) − b(i, j) → ∞,

(3) b(l, j + 1) − a(l, j) = a(l, j) − b(l, j),

(4) M̂\S^{3} × [b(l, n), ∞) ⊂ K_{l} and d(b(l, n), ∂K_{l}) → ∞.

a(l, j) and b(l, j) correspond to bottom and top of each wave respectively. Let us put w(i, j) = w_{b(i,j)} where w_{l} is in 2.A. Also we use the linear trans-

lations T (i, j) : S^{3} × R ↦ S^{3} × R by T (i, j)(m, 0) = (m, b(i, j)). Then we consider the set [u_{l}], u_{l} ∈ B̄^{∗}(K_{l}, w, ϵ)_{0} satisfying the following:

(1)’ u_{l} = φ_{a(l,0)}u_{l} converges to u_{1} ∈ B̄^{∗}(M̂, w, ϵ),

(2)’ [φ_{a(l,j)} − φ_{a(l,j−1)}]u_{l} ∈ B̄^{∗}(K_{l}, w(l, j), ϵ)_{0} and T (l, j)^{∗}([φ_{a(l,j)} − φ_{a(l,j−1)}]u_{l}) converge to u_{j} ∈ B̄^{∗}(S^{3} × R, w, ϵ) for j = 1, …, n.

Then we have a norm as:

|[u_{l}]|B̄^{∗}({K_{l}}_{l}; n) = Sup_{l}||u_{l}||_{l}
≡ Sup_{l}{|φ_{a(l,0)}u_{l}|B̄^{∗}(K_{l}, w, ϵ)_{0}
+ Σ_{j=1}^{n}|[φ_{a(l,j)} − φ_{a(l,j−1)}]u_{l}|B̄^{∗}(K_{l}, w(l, j), ϵ)_{0}}.

The set of [u_{l}] as above forms a Banach space B̄^{∗}({K_{l}}_{l}; n). Again one gets a closed subspace B̄_{0}^{∗}({K_{l}}_{l}; n) similarly as before. Later we will fix the pair of indices {a(l, j)} and {T (i, j)} and will not denote them specifically.

Definition 2.1 Let us fix n ≥ 0. Then we put:

B̂^{∗}(M̂, w, ϵ) = B̂^{∗}(M̂, w, ϵ; n) = B̄^{∗}({K_{l}}_{l}; n)/B̄_{0}^{∗}({K_{l}}_{l}; n).

As before one gets an isomorphism:

\begin{aligned}B̂^{∗}(M̂, ϵ) ≅ B̄^{∗} & (M̂, w, ϵ) ⊕ \\ & B̄^{∗}(S^{3} × R, w, ϵ) ⊕ ⋯ ⊕ (n times) B̄^{∗}(S^{3} × R, w, ϵ). \end{aligned}

Notice the isometric filtration:

B̄^{∗}(M̂, w, ϵ) = B̂^{∗}(M̂, w, ϵ; 0) ⊂ B̂^{∗}(M̂, w, ϵ; 1) ⊂ ⋯ ⊂ B̂^{∗}(M̂, w, ϵ; n).

Let us take a subsequence k(0) < k(1) < …, l << k(l). Then one has another exaustion as K′_{0} ⊂ K′_{1} ⊂ …, K′_{l} = K_{k(l)}. Then we have two kinds of the natural maps:

\begin{matrix}i : B̂^{i}(\{K_{l}\}) ↦ B̂^{i}(\{K′_{l}\}), [u_{l}] ∈ B̄^{i}(K_{l})_{0} → [u_{l}] ∈ B̄^{i}(K′_{l})_{0}, \\ j : B̂^{i}(\{K_{l}\}) ↦ B̂^{i}(\{K′_{l}\}), [u_{l}] ∈ B̄^{i}(K_{l})_{0} → [u_{k(l)}] ∈ B̄^{i}(K′_{l})_{0}. \end{matrix}

Lemma 2.3 Both of i and j give isometric isomorphisms.

P roof : Clearly i is an isometric injection and j is a surjection. Let us see j is isometric. |[u_{l}]| ≥ |j([u_{l}])| follows by definition. Take [u_{l}] ∈ B̂^{i}({K_{l}}) and choose subindices m(l) with lim_{l}||u_{m(l)}||_{m(l)} = |[u_{l}]|. For any small ϵ > 0, there exists l_{0} so that for all l ≥ l_{0}, ||u_{m(l)} − u_{k(l)}|| < ϵ. Thus one gets an estimate:

||u_{k(l)}||_{k(l)} ≥ ||u_{m(l)}||_{m(l)} − ||u_{m(l)} − u_{k(l)}|| ≥ ||u_{m(l)}||_{m(l)} − ϵ.

This verifies that j is also isometric.

Now finally one checks that i is a surjection. For simplicity, one sees it for B̂^{∗}({K_{l}}_{l}; 1). Let us take [u_{l}] ∈ B̂^{∗}({K′_{l}}_{l}; 1) such that φ_{l}u_{l} and T_{2l}^{∗}(1 − φ_{l})u_{l} converge to v_{1} and v_{2} respectively. It is enough to find some [w_{l}] ∈ B̂^{∗}({K_{l}}_{l}; 1) so that φ_{l}w_{l} and T_{2l}^{∗}((1 − φ_{l})w_{l}) both converge to v_{1} and v_{2} respectively.

Let S_{l} : S^{3} × [0, ∞)(⊂ M̂) ≅ S^{3} × [−l, ∞)(⊂ S^{3} × R) be the family of translations, and we take families of cut off functions p_{l} on M̂ and q_{l} on S^{3} × R satisfying Supp p_{l} ⊂ Suppp_{l+1} ⋯ ⊂ M̂ and Supp q_{l} ⊂ Suppq_{l+1} ⋯ ⊂ S^{3} × R. Moreover one may choose these so that Suppp_{l} ⊂ K_{l} and S_{l}^{−1}( Suppq_{l}) ⊂ K_{l}. Then one gets w_{l} = p_{l}v_{1} + S_{2l}^{∗}(q_{l}v_{2}) ∈ B̄^{∗}(K_{l})_{0} and clearly [w_{l}] ∈ B̂^{∗}({K_{l}}_{l}) satisfies the desired condition. This completes the proof.

2.B General setting: Here we have the most general form of the asymptotic Banach spaces. Later we will not use Banach spaces defined in this section. For simplicity, we choose zero weight functions w = w′ = 0.

Let u_{0}, u_{1}, … be a family of u_{l} ∈ B̄^{i}(K_{l})_{0} satisfying:

(1) |u_{l} − u_{l′}|B̄^{i}(K) → 0, l′ ≥ l on every compact subset K ⊂ K_{l} ⊂ K_{l′},

(2) |u_{l}|B̄^{i}(K_{l})_{0} are uniformly bounded.

We put B̄({K_{l}}_{l}) = {[u_{l}]; (1), (2)} as before.

Let B̄u^{∗}({K_{l}}_{l}) be a variant consisting of the set {[u_{l}] : (1)′, (2)} where we replace the topology in (1) by uniform convergence:

(1)′ : |u_{l} − u_{l′}|B̄^{i}(K_{l′})_{0} → 0 uniformly.

B̄u^{∗}({K_{l}}_{l}) is a closed subspace in B̄^{∗}({K_{l}}_{l}). B̄u_{0}^{∗}({K_{l}}_{l}) is defined similarly as before. Then the quotient space B̄^{∗}(M̂) = B̄u^{∗}({K_{l}}_{l})/B̄u_{0}({K_{l}}_{l}) naturally isometrically injects:

I : B̄^{∗}(M̂) ↪ B̂^{∗}({K_{l}}_{l}) = B̄^{∗}({K_{l}}_{l})/B̄u_{0}^{∗}({K_{l}}_{l}).

Lemma 2.4 Any [u_{l}] ∈ B̂^{∗}({K_{l}}_{l}) has a decomposition [u_{l}] = [w_{l}] + [b_{l}] where [w_{l}] ∈ B̄u^{∗}({K_{l}}_{l}) and Suppb_{l} → ∞ as l → ∞.

P roof : By the condition, there is u_{∞} ∈ B̄^{∗}(M̂) such that u_{l} converges to u_{∞} on each compact subset. Let {φ_{l}}_{l} be the previous family of cut off functions. Let us take any decreasing family {δ_{l}}_{l}, δ_{0} >> δ_{1} >> ⋯ → 0. First take l_{0} so that for any l, l′ ≥ l_{0}, |u_{l} − u_{l′}|B̄^{∗}(K_{0}) < δ_{0}. Next take l_{1} ≥ l_{0} so that for any l, l′ ≥ l_{1}, |u_{l} − u_{l′}|B̄^{∗}(K_{1}) < δ_{1}. Similarly we inductively determine l_{2} << l_{3} << . . .. Then we put:

ψ_{i} = φ_{0}, 0 ≤ i ≤ l_{1},
= φ_{1}, l_{1} + 1 ≤ i ≤ l_{2},
= …
= φ_{l}, l_{l} + 1 ≤ i ≤ l_{l+1},
= …

Then we put w_{l} = ψ_{l}u_{l} and b_{l} = (1 − ψ_{l})u_{l}. Clearly this satisfies the desired condition. This completes the proof.

Corollary 2.2 B̂^{i}(M̂) contains B̄^{i}(M̂) ⊕ B̄^{i}(S^{3} × R) ⋯ ⊕ B̄^{i}(S^{3} × R) ⊕ … isometrically.

3 AHS complex

Here we show that the AHS complex between B̂^{∗}({K_{l}}_{l}) is of Fredholm. In order to verify it, we will take two steps. First we consider more simple case. Recall Banach spaces B^{∗}(M̂, w, ϵ) defined in 1.A. Let us fix n ≥ 0 and put:

B^{∗}({K_{l}}_{l}, ϵ) = {[u_{l}] : u_{l} ∈ B^{∗}(K_{0}, ϵ)_{0}, satisfying (1)’, (2)’ in 2.A}.

We have also its closed subspace B_{0}^{∗}({K_{l}}_{l}, ϵ). Then we put:

B̃^{∗}({K_{l}}_{l}, ϵ) ≡ B^{∗}({K_{l}}_{l}, ϵ)/B_{0}^{∗}({K_{l}}_{l}, ϵ).

First we verify Fredholmness of the AHS complex for B̃^{∗}({K_{l}}_{l}), and then for B̂^{∗}({K_{l}}_{l}).

3.B B̃^{∗} case: Now one has the bounded complex:

0 ⟶ B̃^{0}({K_{l}}_{l}) ⟶^{d} B̃^{1}({K_{l}}_{l}) ⟶^{d^{+}} B̃^{2}({K_{l}}_{l}) ⟶ 0  (3.1)

Proposition 3.1 The above complex has closed range. Proof: Let us consider d(B̃^{0}) ⊂ B̃^{1}. Suppose:

|[d(u_{l}^{α})] − [w_{l}]|_{1} → 0, α → ∞ [w_{l}] ∈ B̃^{1}.

We only consider n = 1. The general case is similar. Recall the norm ||u_{l}||_{l}B^{∗} = |φ_{l}u_{l}|B^{∗} + |T_{2l}^{∗}(1 − φ_{l})u_{l}|B^{∗}. Since d : (W_{1}^{4+2ϵ})_{w} ↦ L_{w}^{4+2ϵ}(Λ^{1}) has closed range with null kernel over both M̂ and S^{3} × R, there is a large l_{0} determined by [u_{l}] such that for all l ≥ l_{0}, one gets an estimate (see remark 3.1):

||du_{l}||_{l}B^{1} ≥ C||u_{l}||_{l}B^{0}. ∗

By definition for small ϵ > 0, there is α_{0} with lim_{l→∞}||d(u_{l}^{α}) − w_{l}||_{l}B^{1} < ϵ for α ≥ α_{0}. In particular for each α, there is l(α) such that the set {du_{l(α)}^{α}}_{α} consists of Cauchy sequence in || ||B^{1} topology. Moreover one may assume the estimate ∗ for all u_{l(α)}^{α}. Thus [v_{α} = u_{l(α)}^{α}] defines an element in B^{0}({K_{l(α)}}_{α}). By the condition, [dv_{α} −w_{l(α)}] ∈ B_{0}^{1}({K_{l(α)}}). Combining with the isomorphism j : B̃^{∗}({K_{l}}_{l}) ≅ B̃^{∗}({K_{l(α)}}_{α}) in lemma 2.3, this verifies that d : B̃^{0}({K_{l}}_{l}) ↦ B̃^{1}({K_{l}}_{l}) has closed range.

Next let us consider d^{+}(B̃^{1}) ⊂ B̃^{2}. The following lemma completes the proof of the proposition. Lemma 3.1 There exist a decomposition B̃^{2} ≅ imd^{+} ⊕ H and a bounded map Q : B̃^{2} ↦ B̃^{1} with d^{+}Q| imd^{+} is the identity, where H = H(0) in proposition 1.1. P roof : Recall a decomposition L_{w}^{2+ϵ}(M̂; Λ_{+}^{2}) = imd^{+} ⊕ H where H is of finite dimension, and a bounded map Q with d^{+} ∘ Q = id:

Q : imd^{+} ∩ L_{w}^{2+ϵ}(M̂; Λ_{+}^{2}) ↦ B^{1}(M̂; ϵ).

With respect to the above decomposition, one puts w = w^{1} ⊕ w^{2}, and | |_{0} ≡ | |_{1} ⊕ | |_{2} is the Banach norm on L_{w}^{2+ϵ}(M; Λ_{+}^{2}) where | |_{1} and | |_{2} are the ones on L_{w}^{2+ϵ} and H respectively. Then one has an equivalence C^{−1}| |_{0} ≤ | |_{1} ≤ C| |_{0}.

Let us choose a sequence {k(l)}_{l} such that for any u ∈ H, |u|B^{2} = 1, lim_{l}|(1 − ψ_{l})u|B^{2} = 0 follows for ψ_{l} = φ_{k(l)}. For any element u = [u_{l}] ∈ B̃^{2}, one has a decomposition as φ_{l}u_{l} = u_{l}^{1} ⊕ u_{l}^{2} ∈ B^{2}(M̂, ϵ) where u_{l}^{1} ∈ imd^{+} ⊂ L_{w}^{2+ϵ}(M̂; Λ_{+}^{2}) and u_{l}^{2} ∈ H. With respect to the family {Q(u_{l}^{1})}_{l}, one takes another sequence k(l)′ ≥ k(l) and puts ψ′_{l} = φ_{k(l)′}. Then one may assume |Q(u_{l}^{1}) − ψ′_{l}Q(u_{l}^{1})|B^{1}(M̂, ϵ) → 0. In the Banach space B^{2}({K_{k(l)′}}_{l}), the sequences [v_{l}^{1} = d^{+}(ψ′_{l}Qu_{l}^{1})] and [ψ_{l}u_{l}^{2}] satisfy the condition (1)’ in 2.A, and these defines elements in imd^{+}, H̃ ⊂ B̃^{2}({K_{k(l)′}}_{l}) respectively.

Since d^{+} : B^{1}(S^{3} × R, ϵ) ↦ B^{2}(S^{3} × R, ϵ) is surjective, there is a bounded inverse Q : B^{2}(S^{3} ×R, ϵ) ↦ B^{1}(S^{3} ×R, ϵ) such that d^{+} ∘Q is the identity. Then by choosing a subsequence {k(l)′}_{l} as above, one may assume that the family [w_{l} ≡ (1 − ψ′_{l})(T_{2k(l)′}^{−1})^{∗}Q(T_{2l}^{∗}(1 − φ_{l})u_{l})] gives an element in B̃^{1}({K_{k(l)′}}_{l}).

Thus one assigns [u_{l}] → [d^{+}ψ′_{l}Q(u_{l}^{1}) + d^{+}w_{l}] ⊕ [ψ_{l}u_{l}^{2}]. Passing through the isometric isomorphisms i : B̃^{i}({K_{l}}_{l}) ≅ B̃^{i}({K_{k(l)′}}_{l}) in lemma 2.3, one gets a map B̃^{2}({K_{l}}_{l}) ↦ imd^{+} ⊕ H. Since the norms are invariant under change of indices, this map gives a one to one onto continuous map. This assignment gives the isomorphism.

If [u_{l}^{2}] ∈ B_{0}^{2}, then one puts Q([u_{l}]) = [ψ′_{l}Q(u_{l}) + w_{l}] ∈ B^{1}. This completes the proof.

Next let us compute the cohomology groups. Let us take a subsequence k(0) < k(1) < …, l << k(l). Then one has another exaustion as K′_{0} ⊂ K′_{1} ⊂ …, K′_{l} = K_{k(l)}, and has the corresponding Banach spaces B̃^{i}({K_{l}}) and B̃^{i}({K′_{l}}). As before, since these are all isomorphic, one has also isomorphisms between their cohomology groups.

Recall that the AHS complex:

0 ⟶ B^{0}(M̂, g, ϵ, w) ⟶^{d} B^{1}(M̂, g, ϵ, w) ⟶^{d^{+}} B^{2}(M̂, g, ϵ, w) ⟶ 0 (3.2)

has their cohomology groups H^{0} = H^{1} = 0 and dimH^{2} = b_{2}^{+}(M). Let us consider the corresponding complex:

0 ⟶ B̃^{0}({K_{l}}_{l}) ⟶^{d} B̃^{1}({K_{l}}_{l}) ⟶^{d^{+}} B̃^{2}({K_{l}}_{l}) ⟶ 0.  (3.3)

Here we show that one gets the same result as above. First let us take u = [u_{l}] ≠ 0 ∈ H^{0}. By definition, there is a positive constant C > 0 and a subsequence l with ||u_{l}||_{l}B^{0}(K_{l})_{0} ≥ C and ||d(u_{l})||_{l}L_{2w}^{4+2ϵ}(K_{l})_{0} → 0. This is a contradiction since d : (W_{1}^{4+2ϵ})_{2w} ↦ L_{2w}^{4+2ϵ}(Λ^{1}) has closed range with null kernel. This shows H^{0} = 0.

Next let us choose u = [u_{l}] ∈ H^{1}. Take a sufficiently large l so that ||d^{+}(u_{l})||_{l}L_{w}^{2+ϵ}(K_{l})_{0} is bounded by a small ϵ > 0. Then there is f_{l} ∈ B^{0}(M̂) with ||u_{l} − df_{l}||_{l}B^{1}(M̂) < Cϵ. This can be seen as follows; suppose φ_{l}u_{l} and T_{2l}^{∗}(1 − φ_{l})u_{l} converge to v_{1} and v_{2} in B^{1}(M̂, w, ϵ) and B^{1}(S^{3} × R, w, ϵ) respectively. Then by the assumption, both of v_{i} satisfy d^{+}(v_{i}) = 0. Thus there are f ∈ B^{0}(M̂) and g ∈ B^{0}(S^{3} × R) with df = v_{1} and dg = v_{2}. Using the notations in the proof of lemma 2.3, one gets [f_{l} = p_{l}f + S_{2l}^{∗}(q_{l}g)] ∈ B^{0}({K_{l}}_{l}). Clearly [u_{l}] = [df_{l}] ∈ B^{1}({K_{l}}_{l}). This shows H^{1} = 0.

Let us denote H^{2} = H^{2}(B̃^{∗}({K_{l}}_{l})) and similarly H^{2}(B^{∗}(M̂)). Then there is a natural map (using a family of cut off functions {φ_{l}}):

I : H = H^{2}(B^{∗}(M̂)) ↦ H^{2}(B̃^{∗}({K_{l}}_{l}))

by I(u) = [φ_{l}u] ≡ [u_{l}]. We see that this gives an isomorphism. Suppose I(u) = 0. Then there is [v_{l}] ∈ B^{1}({K_{l}}_{l}) with [u_{l} − d^{+}(v_{l})] ∈ B_{0}^{2}({K_{l}}_{l}). In particular for any small δ > 0, there is a large l with |u_{l} − d^{+}(v_{l})|L_{w}^{2+ϵ} < δ. Since d^{+} : B^{1}(M̂) ↦ B^{2}(M̂) has closed range, this shows u = 0 ∈ H. Thus I is an injection. Surjectivity follows from lemma 3.1. This shows dimH^{2}(B̃^{∗}) = b_{+}^{2}(M).

3.C ϵ collapsed Banach spaces: Here we define more general asymptotic Banach spaces B̂^{∗}({K_{l}, ϵ_{l}}_{l}), and in the next section we study the AHS complex over B̂^{∗}({K_{l}, ϵ_{l}}_{l}).

Let K_{0} ⊂ K_{1} ⊂ ⋯ ⊂ M̂ be an exaustion where (M̂, g) is the cylindrical manifold. One may assume that (M̂\K_{0}, g) is isometric to (S^{3} × [0, ∞), g|S^{3} + dt^{2}). Let us choose a family of small and positive constants:

1 >> ϵ_{0} ≥ ϵ_{1} ≥ ⋯ > 0.

Then we have Banach spaces L̂_{w}^{p}(K_{l}) = L̂_{w}^{p}(K_{l}; ϵ_{l}) (L̂_{w}^{p}(K_{l})_{0}). L̂_{2w}^{4} is defined by the norm:

|u|L̂_{2w}^{4}(K_{l}, ϵ_{l}) = sup_{0≤ϵ≤ϵ_{l}}{|u|L_{2w}^{4+2ϵ}(K_{l})}.

L̂_{w}^{2} is also similarly defined (see 2.A). Using these, one obtains similar Banach spaces:

B̄^{0}(K_{l}, ϵ_{l})_{0} = {u ∈ L̂_{2w′}^{4}(K_{l}, ϵ_{l})_{0}, du ∈ L̂_{2w}^{4}(K_{l}, ϵ_{l})_{0}},
B̄^{1}(K_{l}, ϵ_{l})_{0} = {u ∈ L̂_{2w}^{4}(K_{l}, ϵ_{l})_{0}, d(u) ∈ L̂_{w}^{2}(K_{l}, ϵ_{l})_{0}},
B̄^{2}(K_{l}, ϵ_{l})_{0} = L̂_{w}^{2}(K_{l}, ϵ_{l}; Λ_{+}^{2})_{0}.

We recall the norms || ||_{l} introduced in 2.A. If one chooses a subfamily {ϵ′_{l} = ϵ_{k(l)}}_{l}, l ≤ k(l), then the bound holds:

||u||_{l}B̄^{i}(K_{l}, ϵ′_{l})_{0} ≤ ||u||_{l}B̄^{i}(K_{l}, ϵ_{l})_{0}.

In order to define asymptotic Banach spaces, we use the formulation used in 2.A. Let us define:

B̄^{i}({K_{l}, ϵ_{l}}_{l}) = {[u_{l}] : u_{l} ∈ B̄^{i}(K_{l}, ϵ_{l})_{0}, (1), (2) below}

with the norm |[u_{l}]| = Sup_{l}||u_{l}||_{l}B̄^{i}(K_{l}, ϵ_{l})_{0}. Let {a(i, j)} and {T (i, j)} be the families of indices and translations in 2.A. Here [u_{l} ∈ B̄^{i}(K_{l})_{0}] is a sequence satisfying:

(1) |φ_{l}u_{l} − φ_{l′}u_{l′}|B̄^{i}(K_{l′}, w, ϵ_{l′})_{0} → 0 as l, l′ → ∞, l′ ≥ l,

(2) u(l, j) ≡ [φ_{a(l,j)} − φ_{a(l,j−1)}]u_{l} ∈ B^{∗}(K_{l}, w(l, j), ϵ_{l})_{0} and |T (l, j)^{∗}u(l, j) − T (l′, j)^{∗}u(l′, j)| ∈ B̄^{∗}(S^{3} × R, w, ϵ_{l′}) → 0, l′ ≥ l → ∞.

As before one has the closed subspace B̄_{0}^{i}({K_{l}, ϵ_{l}}_{l}) ⊂ B̄^{i}({K_{l}, ϵ_{l}}_{l}). When we take the quotient spaces, we divide into two cases; suppose all ϵ_{l} are the same ϵ > 0. Then we put the quotient space by:

B̂^{i}({K_{l}}_{l}) = B̄^{i}({K_{l}}_{l})/B̄_{0}^{i}({K_{l}}_{l}).

In this case lemma 2.3 tells us that B̂^{∗}({K_{l}}_{l}) doesnot depend on choice of partitions {K_{l}}_{l}, and we denote them as B̂^{∗}(M̂).

Let us consider the case, strictly decreasing constants ϵ_{l} > ϵ_{l+1} > ⋯ → 0. If we take another partition {K′_{l}}_{l} with K_{l} ⊂ K′_{l}, then we have a natural embedding B̂^{∗}({K_{l}, ϵ_{l}}_{l}) ⊂ B̂^{∗}({K′_{l}, ϵ_{l}}_{l}). Then we take a direct limit and denote as:

B̂^{∗}(M̂) = limB̂^{∗}({K_{l}}_{l}).

3.D AHS complex for B̄^{∗}({K_{l}, ϵ_{l}}_{l}): In this section we consider the AHS complex over B̂^{∗}(M̂, g). When we define B̂^{∗}(M̂, g), we have used a family of non increasing constants {ϵ_{l}}_{l}. We will study analytic property of the AHS complex {B̂^{∗}(M̂, g), d^{∗}} including the case when the family of constants approaches to zero, ϵ_{0} >> ϵ_{1} >> … ϵ_{l} >> ⋯ → 0. In particular we show that it is of Fredholm and the cohomlogy groups are the expected ones.

We need the following preparation. Let us take pairs (p, p′) ∈ {(2, 2+ϵ), (4, 4+ 2ϵ)} and Q be a positive second order self adjoint elliptic operator over S^{3}. Let us put the Banach spaces:

L̇^{p} = Max {L^{p}, L^{p′}}.

Namely we have |u|L̇^{p} = max {|u|L^{p}, |u|L^{p′}}.

When using L̇^{4} spaces, we restrict the following case; Q = Δ_{0} + δ > 0 : Ẇ_{2}^{4}(S^{3}) ↦ L̇^{4}(S^{3}) where Δ_{0} is the Laplace operator over functions.

Proposition 3.2

Δ = −\frac{d^{2}}{dt^{2}} + Q : Ẇ_{2}^{p}(S^{3} × R) ↦ L̇^{p}(S^{3} × R)

gives an isomorphism with a uniform bound:

|Δ(u)|L̇^{p}(S^{3} × R) ≥ C|u|Ẇ_{2}^{p}(S^{3} × R)

where C is independent of choice of small ϵ ≥ 0.

Sublemma 3.1

Δ : W_{2}^{p}(S^{3} × R) ↦ L^{p}(S^{3} × R)

is an isomorphism.

P roof : For p = 2, the result is well known, using the exponential decay estimate for vectors with low spectrum.

Let us consider p = 4. Let us take u ∈ W_{2}^{4}(S^{3} × R) and put u_{t} ∈ W_{2}^{4}(S^{3}) a.e.. Notice that by the Sobolev embedding, u is a continuous section. Suppose |Δ(u)|L^{4} is small. We put f(t) = ∫_{S^{3}} |u_{t}|^{4}vol. Then we have estimates:

f(t)′ = 4 ∫ < u′, u > |u|^{2},
f′′ = ∫ < u′′, u > |u|^{2} + ∫ |u′|^{2}|u|^{2} + 2 ∫ | < u′, u > |^{2}
≥ ∫ < Qu, u > |u|^{2} − μf

where δ > μ ≥ 0. Here we use the condition Q = Δ_{0}+δ. We have the following:

\begin{matrix}< Δ_{0}u_{t}, u_{t} > |u_{t}|^{2}vol = − ∗ d ∗ du_{t}u_{t}^{3}vol = −(d ∗ du_{t})u_{t}^{3}, \\ ∫ < Δ_{0}u_{t}, u_{t} > u_{t}^{2} = ∫ 3u_{t}^{2}du_{t} ∧ ∗du_{t} = 3 ∫ |du_{t}|^{2}u_{t}^{2} ≥ 0. \end{matrix}

Thus we have an estimate f′′ ≥ (δ − μ)f. The rest follows from the ordinary method. Namely one can verify the exponential decay for f and Fredholmness for Δ. This completes the proof.

Proof of proposition: First we show that there exists a constant C independent of ϵ with a bound:

|Δ(u)|L̇^{p}(S^{3} × R) ≥ C|u|L̇^{p}(S^{3} × R).

Let us take u with |u|L̇^{p}(S^{3} × R) = 1. When |u|L̇^{p}(S^{3} × R) = |u|L^{p}(S^{3} × R), then the result follows from the above sublemma.

Suppose |u|L̇^{2}(S^{3} × R) = |u|L^{2+ϵ}(S^{3} × R). We decompose:

|u|^{2+ϵ}L^{2+ϵ}(S^{3} × R) = Σ_{n}|u|^{2+ϵ}L^{2+ϵ}(S^{3} × [n, n + 1]).

Then by the Sobolev inequality, there is a constant C independent of ϵ and n with a bound:

|u|L^{2+ϵ}(S^{3} × [n, n + 1]) ≤ C|u|W_{1}^{2}(S^{3} × [n, n + 1]).

Thus one gets an estimate:

Σ_{n}|u|^{2}L^{2+ϵ}(S^{3} × [n, n + 1]) ≤ CΣ_{n}|u|^{2}W_{1}^{2}(S^{3} × [n, n + 1]) = C|u|^{2}W_{1}^{2}(S^{3} × R).

Here we claim the inequality:

Σ_{n}|u|^{2+ϵ}L^{2+ϵ}(S^{3} × [n, n + 1]) ≤ [Σ_{n}|u|^{2}L^{2+ϵ}(S^{3} × [n, n + 1])]^{\frac{2+ϵ}{2}}.

¿From this, we get the estimate for p = 2:

|u|L^{2+ϵ}(S^{3} × R) ≤ C|u|W_{1}^{2}(S^{3} × R).

In fact let us put b_{n} = |u|^{2}L^{2+ϵ}(S^{3} × [n, n + 1]) ≤ 1 and \frac{2+ϵ}{2} = 1 + δ. Then we show the inequality Σ_{n}b_{n}^{1+δ} ≤ (Σ_{n}b_{n})^{1+δ}. By normalizing one may assume Σ_{n}b_{n} = 1. Since b_{n} ≤ 1, clearly we have Σ_{n}b_{n}^{1+δ} ≤ Σ_{n}b_{n}. Thus we have the estimate:

Σ_{n}b_{n}^{1+δ} ≤ Σ_{n}b_{n} = (Σ_{n}b_{n})^{1+δ}.

Now we have the desired inequalities:

C|Δ(u)|L^{2} ≥ |u|W_{2}^{2} ≥ |u|W_{1}^{2} ≥ C′|u|L^{2+ϵ}.

This implies the estimate |Δ(u)|L̇^{2} ≥ C|u|L̇^{2}.

For p′ = 4, the method is parallel. Here we note the Sobolev embedding W_{1}^{4} ↪ L^{4+2ϵ}.

Now we want to obtain the inequality |Δ(u)|L̇^{p} ≥ C|u|Ẇ_{2}^{p}. By the elliptic estimate, we have:

|u|W_{2}^{4+2ϵ} ≤ C\{|u|^{2}W_{1}^{4+2ϵ} + |Δ(u)|^{2}L^{4+2ϵ}\}^{\frac{1}{2}}.

Suppose |u_{i}|Ẇ_{2}^{p} = 1 and |Δ(u_{i})|L̇^{p} → 0. If |u_{i}|Ẇ_{2}^{p} = |u_{i}|W_{2}^{p}, this contradicts to |Δ(u_{i})|L^{p} → 0 and the above sublemma. Thus let |u_{i}|Ẇ_{2}^{p} = |u_{i}|W_{2}^{4+2ϵ}. Combining with |Δ(u_{i})|L^{4+2ϵ} → 0 and the above elliptic estimate, we have 1 − δ ≤ |u_{i}|W_{1}^{4+2ϵ} ≤ 1. Using the Sobolev embeddings:

W_{2}^{2} ↪ W_{1}^{2+ϵ}, W_{2}^{4} ↪ W_{1}^{4+2ϵ}

and a similar argument as above, we get the estimate 1 − δ ≤ |u_{i}|W_{1}^{4+2ϵ} ≤ C|u_{i}|W_{2}^{4}. Combining with the estimate |u_{i}|W_{2}^{4} ≤ C|Δ(u_{i})|L^{4}, this is a contradiction. Thus we get the desired estimate:

|Δ(u_{i})|L̇^{p} ≥ C|u_{i}|Ẇ_{2}^{p}.

This completes the proof.

Corollary 3.1 There exists Q : L̂_{w}^{2}(S^{3} × R) ↦ (Ŵ_{1}^{2})_{w} ∩ Kerd_{w}^{∗} with d^{+}Q = id.

P roof : We proceed as in the proof of proposition 1.1. Let us put ∇ = d^{+}⊕d_{w}^{∗} : (Ŵ_{1}^{2})_{w} ↦ L̂_{w}^{2}. Then we can see that there is a constant C > 0 with |∇| ≥ C.

We can also see Δ : (Ŵ_{2}^{2})_{w} ↦ L̂_{w}^{2} satisfies |Δ| ≥ C > 0 on functions. Thus (Ŵ_{1}^{2})_{w} decomposes as:

(Ŵ_{1}^{2})_{w} ≅ imd ⊕ Kerd_{w}^{∗}

by u → dΔ^{−1}d_{w}^{∗}u + u − dΔ^{−1}d_{w}^{∗}u. Since |∇| ≥ C > 0, the bounded map d^{+} : Kerd_{w}^{∗} ↦ L̂_{w}^{2} satisfies |d^{+}| ≥ C > 0. Now we put Qd^{+}(v) = v − dΔ^{−1}d_{w}^{∗}v. This completes the proof.

¿From the estimate at the end of senction 1, one gets the following map:

Q : B̄^{2}(S^{3} × R) ↦ B̄^{1}(S^{3} × R) ∩ Kerd_{w}^{∗}

with d^{+}Q = id.

Remark 3.1: Recall that since Δ over M̂ can be written as d_{w}^{∗} ∘ d, it follows from the above that d : (Ŵ_{2}^{4})_{2w} ↦ (Ŵ_{1}^{4})_{2w} has closed range.

Let us consider d : (Ŵ_{1}^{4})_{2w} ↦ L̂_{2w}^{4}. This has also closed range as follows. ¿From sublemma 3.1, d : (W_{1}^{2})_{2w} ↦ L_{2w}^{2} has closed range. Then we have an estimate:

|u|^{4}L_{2w}^{4} = ∫ exp(2w)u^{4} ≤ C2 ∫ exp(2w)u^{2}|du|^{2} ≤ C|u|^{2}L_{2w}^{4}|du|^{2}L_{2w}^{4}.

Thus we get a bound |u|L_{2w}^{4} ≤ C|du|L_{2w}^{4}. Then following the proof of proposition 3.2, we get the desired estimate |u|L̂_{2w}^{4} ≤ C|du|L̂_{2w}^{4}.

Lemma 3.2 Let us take δ and ϵ_{0}. Then for 0 < δ′ = δ′(δ) < δ, 0 ≤ ϵ_{l} ≤ ϵ_{0}, the AHS complex:

0 ⟶ B̂^{0}(M̂) ⟶^{d} B̂^{1}(M̂) ⟶^{d^{+}} B̂^{2}(M̂) ⟶ 0  (3.4)

has closed range, where B̂^{∗}(M̂) = B̂^{∗}({K_{l}, ϵ_{l}}_{l}, w(′)).

P roof : Recall that for 0 ≤ ϵ ≤ ϵ_{0}, one gets an isomorphism:

d : B^{0}(M̂) ≅ imd ⊂ B^{1}(M̂).

Let | |_{i}^{ϵ}, i = 0, 1, be the right and left hand sides norms respectively. Then one has equivalences C_{ϵ}^{−1}| |_{0}^{ϵ} ≤ | |_{1}^{ϵ} ≤ C_{ϵ}| |_{0}^{ϵ}.

Sublemma 3.2 For d : B^{0}(M̂; ϵ) ↦ B^{1}(M̂; ϵ), there is a uniform bound of the family of constants C^{−1} ≤ C_{ϵ} ≤ C.

P roof : We show a uniform bound |f|B^{0}(M̂; ϵ) ≤ C|df|B^{1}(M̂; ϵ) for all 0 ≤ ϵ ≤ ϵ_{0}, which follows from another one |f|L_{2w′}^{4+2ϵ}(M̂) ≤ C|df|L_{2w}^{4+2ϵ}(M̂). Let us choose δ′ < δ which will be determined later, and take the corresponding w′ with weight δ′. Then there is a constant C independent of ϵ with a bound:

| |L_{2w′}^{4}(M̂) ≤ C| |L_{2w}^{4+2ϵ}(M̂).

This follows from the local Holder estimate L_{loc}^{4} ↪ L_{loc}^{4+2ϵ} and a local to global method (see below, or proof of proposition 4.1). Then we show another estimate |u|L_{2w′}^{4} ≤ C|du|L_{2w}^{4}. Using Hilbert space structure, one gets easily an estimate |u|(W_{1}^{2})_{w} ≤ C|du|L_{w}^{2} (see [K] 6.A). Then we will have inequalities:

|u|L_{2w′}^{4} ≤ C|u|(W_{1}^{2})_{w′} ≤ C|du|L_{w′}^{2} ≤ |du|L_{2w}^{4}.

Notice local Sobolev embedding (W_{1}^{2})_{loc} ↪ L_{loc}^{4}. Then the first and the last inequalities follow from the next estimates respectively:

Σ_{n}a_{n}^{4}exp(2nδ) ≤ (Σ_{n}a_{n}^{2}exp(nδ))^{2},

\begin{aligned}Σ_{n}a_{n}^{2}exp(nδ′) & = Σ_{n}[a_{n}^{2}exp(nδ)][exp(n(δ′ − δ))] \\ & ≤ C(Σ_{n}a_{n}^{4}exp(2nδ))^{\frac{1}{2}}. \end{aligned}

Finally we show a uniform bound |f|L_{2w}^{4+2ϵ} ≤ C|f|(W_{1}^{4})_{2w}. Notice local Sobolev embedding |f|L_{loc}^{4+2ϵ} ≤ C|f|(W_{1}^{4})_{loc}. Using this constant C, one proceeds a similar argument as before. By normalizing one may assume |f|(W_{1}^{4})_{2w} = C^{−1} < 1. In particular for each n, it follows a_{n} ≡ |f|L^{4+2ϵ}(S^{3} × [n, n + 1]) ≤ Cb_{n} ≡ |f|W_{1}^{4}(S^{3} × [n, n + 1]) ≤ 1. Let us put 1 − μ = \frac{4}{4+2ϵ}. Then one gets the estimate:

|f|L_{2w}^{4+2ϵ} = (Σ_{n}a_{n}^{4+2ϵ}exp(2nδ))^{(4+2ϵ)^{−1}}
≤ (Σ_{n}a_{n}^{4}exp(2nδ))^{(4+2ϵ)^{−1}}
≤ C(Σ_{n}b_{n}^{4}exp(2nδ))^{(4+2ϵ)^{−1}} = C|f|^{1−μ}(W_{1}^{4})_{2w}
= C|f|(W_{1}^{4})_{2w}|f|^{−μ}(W_{1}^{4})_{2w} ≤ C′|f|(W_{1}^{4})_{2w}.

This verifies the claim.

Now suppose a family f_{i} ϵ_{i} → 0, satisfies |f_{i}|L_{2w′}^{4+2ϵ_{i}} = 1 and |df_{i}|L_{2w}^{4+2ϵ_{i}} → 0. Then one gets the estimate:

|f_{i}|L_{2w′}^{4+2ϵ} ≤ C|f_{i}|(W_{1}^{4})_{2w′}
≤ C|f_{i}|L_{2w′}^{4} + Cδ_{i}
≤ C|df_{i}|L_{2w′′}^{4} + Cδ_{i} ≤ C|df_{i}|L_{2w}^{4+2ϵ_{i}} + Cδ_{i} → 0.

This is a contradiction. This completes the proof.

Proof of lemma: Suppose there exists a family {[u_{l}^{α}]}_{α} ⊂ B̂^{0}({K_{l}, ϵ_{l}}_{l}) with |[u_{l}^{α}]|B̂^{0} = 1 and lim_{α}|[du_{l}^{α}]|B̂^{1} → 0. Let us take a decreasing family δ_{α} and assume lim sup_{l}||du_{l}^{α}||_{l}B̄^{1} < δ_{α}, while lim sup_{l}||u_{l}^{α}||_{l}B̄^{1} = 1. In particular for some arbitrarily large l = l(α), one has the estimates:

||du_{l}^{α}||_{l}B̄^{1} ≤ δ_{α}, ||u_{l}^{α}||_{l}B̄^{0} ≥ 1 − δ_{α}.

Then one gets the inequality:

1 − δ_{α} ≤ ||u_{l}^{α}||_{l}B̄^{0}(K_{l}, ϵ_{l})_{0} ≤ C||du_{l}^{α}||_{l}B̄^{1}(K_{l}, ϵ_{l})_{0} ≤ Cδ_{α}.

This is a contradiction. Thus d : B̂^{0}({K_{l}, ϵ_{l}}_{l}) ↦ B̂^{1}({K_{l}, ϵ_{l}}_{l}) has closed range.

Now we have another method to verify that d has closed range which also works for d^{+}.

Let us take any subindices {k(l)}_{l} and consider the restriction map:

\begin{matrix}j : B̂^{∗}(\{K_{l}, ϵ_{l}\}_{l}) ↦ B̂^{∗}(\{K_{k(l)}, ϵ_{k(l)}\}_{l}), \\ j_{0} : B̄^{∗}(\{K_{l}, ϵ_{l}\}_{l}) ↦ B̄^{∗}(\{K_{k(l)}, ϵ_{k(l)}\}_{l}). \end{matrix}

We claim that j gives an isometric isomorphism between Banach spaces. Since j_{0} is surjective, j is also so. Let us take [u_{l}] ∈ B̂^{∗}({K_{l}, ϵ_{l}}_{l}). We choose subindices {m(l)} and {k(l)′} ⊂ {k(l)} so that lim_{l}||u_{m(l)}|| = |[u_{l}]|B̂^{∗}({K_{l}, ϵ_{l}}_{l}) and lim_{l}||u_{k(l)′}|| = |j([u_{l}])|B̂^{∗}({K_{k(l)}, ϵ_{k(l)}}_{l}). By the definition, for any small μ > 0, there is a large l_{0} such that for all l′ ≥ l ≥ l_{0}, ||u_{l′} − u_{l}||B̄^{∗}(K_{l′}, ϵ_{l′}) ≤ μ. In particular one gets ||u_{k(l)′} − u_{m(l′)}||B̄^{∗}(K_{m(l′)}, ϵ_{m(l′)}) ≤ μ. Then we have estimates:

− ||u_{k(l)′} − u_{m(l′)}||B̄^{∗}(K_{m(l′)}, ϵ_{m(l′)}) + ||u_{m(l′)}||B̄^{∗}(K_{m(l′)}, ϵ_{m(l′)})
≤ ||u_{k(l)′}||B̄^{∗}(K_{m(l′)}, ϵ_{m(l′)})
≤ ||u_{k(l)′} − u_{m(l′)}||B̄^{∗}(K_{m(l′)}, ϵ_{m(l′)}) + ||u_{m(l′)}||B̄^{∗}(K_{m(l′)}, ϵ_{m(l′)}).

Thus we get the estimate:

\begin{matrix}|[u_{l}]|B̂^{∗}(\{K_{l}, ϵ_{l}\}_{l}) = lim_{l}||u_{m(l′)}||B̄^{∗}(K_{m(l′)}, ϵ_{m(l′)}) \\ ≤ lim_{l}||u_{k(l)′}||B̄^{∗}(K_{k(l)′}, ϵ_{k(l)′}) = |j([u_{l}])|B̂^{∗}(\{K_{k(l)}, ϵ_{k(l)}\}_{l}). \end{matrix}

Since j is distance decreasing, this verifies the claim.

Let us take a compactly supported cut off function φ_{0} : M̂ ↦ [0, 1] with φ_{0}|K_{0} ≡ 1. One may assume (M̂\K_{0}, g) is isometric to (S^{3} ×[0, ∞), g|S^{3} +dt^{2}). Let us consider d : B̂^{0}({K_{l}, ϵ_{l}}_{l}) ↦ B̂^{1}({K_{l}, ϵ_{l}}_{l}). By the above sublemma, one has a uniform bound:

C||d(1 − φ_{0})u_{l}||B̄^{1}(K_{l}; ϵ_{l})_{0} ≥ ||(1 − φ_{0})u_{l}||B̄^{0}(K_{l}; ϵ_{l})_{0}.

Let us take a family [u_{l}^{α}] ∈ B̄^{0} with |[u_{l}^{α}]|B̄^{0} = 1, |[du_{l}^{α}]|B̄^{1}({K_{l}, ϵ_{l}}_{l}) → 0 as α → ∞. We show that:

d : B̄^{0}({K_{l}, ϵ_{l}}_{l}) ↦ B̄^{1}({K_{l}, ϵ_{l}}_{l}) ∗

has closed range asymptotically in the following sense; there exists a subindices k(l, α) and a(α) with {[φ_{0}u_{k(l,α)}^{a(α)}]}_{α} convergent in L^{4+2ϵ_{0}}(K_{0})_{0}. Here is the inclusion … {k(l, m)}_{l} ⊂ … {k(l, 1)}_{l} ⊂ {l}_{l}. Notice that {[du_{k(l,α)}^{a(α)}]}_{α} also converges in B̄^{1}.

We have estimates:

|[u_{l}^{α}]|B̄^{0} ≤ |[φ_{0}u_{l}^{α}]|B̄^{0} + |[(1 − φ_{0})u_{l}^{α}]|B̄^{0}
≤ |[φ_{0}u_{l}^{α}]|B̄^{0} + C|[d(1 − φ_{0})u_{l}^{α}]|B̄^{1}
≤ |[φ_{0}u_{l}^{α}]|B̄^{0} + C|[du_{l}^{α}]|B̄^{1} + C|[dφ_{0} ∧ u_{l}^{α}]|B̄^{1}
≤ |[φ_{0}u_{l}^{α}]|B̄^{0} + C|[du_{l}^{α}]|B̄^{1} + C|[u_{l}^{α}]|B̄^{1}({K_{l} ∩ Supp dφ_{0}}_{l})
≤ C|[φ′_{0}u_{l}^{α}]|B̄^{0} + C|[du_{l}^{α}]|B̄^{1}

where Supp φ_{0} ⊂ Supp φ′_{0}. Let us put ϵ = ϵ_{0}. Then φ_{0}u_{l}^{α} ∈ L^{4+2ϵ}(K_{0})_{0} for all α, l by the compact embedding W_{1}^{4}(K_{0}) ↪ L̂^{4}(K_{0}). Thus it is enough to see that {[φ_{0}u_{k(l,α)}^{a(α)}]}_{α} ⊂ B̄^{0}(K_{0})_{0} ⊂ L^{4+2ϵ}(K_{0})_{0} is a convergent sequence.

Let us put v_{l}^{α} = φ_{0}u_{l}^{α} and choose any decreasing sequence δ_{i}(> 0) → 0. Since the sets {v_{l}^{α}}_{l} consists of Cauchy sequences in W_{1}^{4}(K_{0})_{0}, they converge to u^{α} ∈ W_{1}^{4}(K_{0})_{0} for each α. First choose a subsequence a(i) with |u^{a(i)} − u^{a(j)}|L^{4+2ϵ}(K_{0})_{0} < max{δ_{i}, δ_{j}} for all i, j. Then we put w_{α}^{l} = v_{α}^{a(l)}, w^{l} = u^{a(l)}.

Next let us choose subindices α(1, l) with |w_{α(1,i)}^{1} − w^{1}|W_{1}^{4}(K_{0}) < δ_{1} and |du_{α(1,i)}^{a(1)}|L_{loc}^{4+ϵ}(K_{0}) < δ_{1} for all i. Then choose subindices {α(2, i)}_{i} ⊂ {α(1, i)}_{i} with |w_{α(2,i)}^{2} − w^{2}|W_{1}^{4}(K_{0}) < δ_{2} and |du_{α(2,i)}^{a(2)}|L^{4+2ϵ}(K_{0}) < δ_{2}. We inductively choose α(l, i) as above. Thus we get the family {[z_{l}^{a} = w_{α(a,l)}^{a}]}_{l}, and the estimate:

\begin{aligned}|z_{l}^{i} & − z_{l}^{j}|L^{4+2ϵ}(K_{0})_{0} ≤ |z_{l}^{i} − w^{i}|L^{4+2ϵ}(K_{0})_{0} \\ & + |w^{i} − w^{j}|L^{4+2ϵ}(K_{0})_{0} + |w^{j} − z_{l}^{j}|L^{4+2ϵ}(K_{0})_{0} ≤ δ_{i} + δ_{i} + δ_{j}. \end{aligned}

We put α(i, l) = k(l, i). Then this shows that {[z_{l}^{i} = φ_{0}u_{k(l,i)}^{a(i)}]}_{l} is a convergent sequence. Notice the inclusion {k(l, i)}_{l} ⊂ {k(l, i − 1)}_{l}.

Now let us consider d : B̂^{0} ↦ B̂^{1}. Suppose a family {[u_{l}^{α}]}_{α} ⊂ B̂^{0} satisfies |[u_{l}^{α}]|B̂^{0} = 1 and |[du_{l}^{α}]|B̂^{1} → 0. Thus there are arbitrarily large l = l(α) satisfying ||du_{l}^{α}||_{l}B̄^{1}(K_{l}) ≤ δ_{α} and ||u_{l}^{α}||_{l}B̄^{0}(K_{l}) ≥ 1 − δ_{α}.

We put f_{l} = φ_{0}u_{k(l,l)}^{a(l)} = z_{l}^{l} and restrict {[z_{l}^{α} = φ_{0}u_{k(l,l)}^{α}]_{l}}_{α} ⊂ B̂^{0}({K_{k(l,l)}}_{l}) ⊂ B̂^{0}({K_{l}}_{l}). Then we claim that the family converges to [f_{l}] ∈ B̂^{0}({K_{k(l,l)}}_{l}).

Since we consider B̂^{∗} norm, one may assume α, β ≤ l, l′. By choice, we have conditions:

|z_{l}^{α} − z_{l′}^{α}|B̄^{0} < 2δ_{α}, |z_{l}^{α} − z_{l}^{β}|B̄^{0} < 3 max(δ_{α}, δ_{β}).

Thus we get the claim. This verifies closedness of d.

Let us consider d^{+}. Let us take [u_{l}^{α}] ∈ B̂^{1} satisfying |[u_{l}]|B̂^{1} = 1 and |[du_{l}^{α}] − [w_{l}]|B̂^{2} → 0 as α → ∞. Notice that the Laplace operator Δ : W_{2}^{2+ϵ}(S^{3}; Λ^{1}) ↦ L^{2+ϵ}(S^{3}; Λ^{1}) admits a uniform bound |Δ| ≥ C where C is independent of ϵ. In particular d^{+} : (Ŵ_{1}^{2})_{w}(S^{3} ×R)∩ Kerd_{w}^{∗} ↦ L̂_{w}^{2}(S^{3} ×R) admits a uniform bound |d^{+}| ≥ C.

Let us fix a smooth Riemannian metric h on M such that there is an isometry (M̂ ∩ Suppφ_{0}, g) ≅ (M̄ ⊂ M, h) where M̂ = M̄ ∪_{S^{3}} S^{3} × [0, ∞). Then with respect to h, one constructs another Banach spaces B̂^{∗}(M, h, {ϵ_{l}}_{l}).

Sublemma 3.3 Its cohomology groups are isomorphic to the de Rham’s one.

In fact one can check Poincaré lemma.

Proof of lemma: Thus there is Q_{1} : im d^{+} ⊂ B̂^{2}(M, h) ↦ Kerd^{∗} ⊂ B̂^{1}(M, h) with d^{+} ∘ Q_{1} = id. Let (S^{3} × R, g = g|S^{3} + dt^{2}) be the cylinder. Then there is also Q_{2} : B̂^{2}(S^{3} × R) ↦ B̂^{1}(S^{3} × R) with d^{+} ∘ Q_{2} = id.

Let I : H(0) ⊂ B̂^{2}(M̂) be as before. Then using a cut off function ψ, one may represent any element u ∈ H^{2} by a compactly supported form, u′ = u − d^{+}(ψQ_{2}(ψu)) ∈ H(0)′. These forms consists of H^{2}(B̂^{∗}(M, h), d_{∗}). Let us

put ψ = φ0^{\frac{1}{2}}, ψ′ = (1 − φ_{0})^{\frac{1}{2}} and:

Q′ = ψQ_{1}ψ + ψ′Q_{2}ψ′ : imd^{+} ⊂ B̂^{2}(M̂) ↦ B̂^{1}(M̂).

We follow the argument in [DS] p212.

In corollary 3.1, we have seen that for any u = [u_{l}] ∈ image Q_{2}, each u_{l} lies in Kerd_{w}^{∗}. In particular u_{l} ∈ (W_{1}^{2})_{loc}. Thus from compact embedding (W_{1}^{2})_{loc} ↪ L_{loc}^{2+ϵ}, it follows d^{+} ∘ Q′ − id : imd^{+} ⊂ B̂^{2}(M̂) ↦ im d^{+} ⊂ B̂^{2}(M̂) is compact.

Let w_{1}, …, w_{l} and d^{+}(α_{1}), …, d^{+}(α_{l}) be the kernel and cokernel sets for d^{+}Q′. Then we put:

Q = (Q′ + Σ_{i=1}^{l}(w_{i}, )α_{i})T^{−1} : imd^{+} ⊂ B̂^{2}(M̂) ↦ B̂^{1}(M̂)

where T = d^{+} ∘ Q′ + Σ_{i=1}^{l}(w_{i}, )d^{+}(α_{i}).

Q satisfies d^{+} ∘ Q = id. This implies the existence of a bounded map:

Q : imd^{+} ⊂ B̂^{2}(M̂) ↦ B̂^{1}(M̂)

with d^{+} ∘ Q = id. Thus d^{+} has closed range. This completes the proof.

Proposition 3.3 The AHS complex between B̂^{i}(M̂) has its cohomology groups as:

H^{0}(B̂^{∗}({K_{l}, ϵ_{l}}_{l}) = 0, H^{1}(B̂^{∗}({K_{l}, ϵ_{l}}_{l}) = 0, H^{2}(B̂^{∗}({K_{l}, ϵ_{l}}_{l}) ≅ H = H(0).

In particular dimH^{2} = b_{+}^{2}(M).

P roof : H^{0} = 0 follows from combination of the proof of proposition 1.1 and 3.B.

Next we claim that there exists Q = Q(M̂) : L̂^{2}(M̂, ϵ′)∩ imd^{+} ↦ B̄^{1}(M̂, ϵ′) with d^{+} ∘ Q = id and |Q| ≤ C where C is independent of 0 ≤ ϵ′ ≤ ϵ. Assuming this for the moment, we proceed as follows. For H^{2}, it also follows from 3.B and existence of Q above.

Let us consider H^{1}. For simplicity we consider only the case n = 1. Then we take [α_{l}] ∈ H^{1}. Thus lim||d^{+}(α_{l})||_{l} = 0. Then since φ_{l}α_{l} − Q(M̂)(d^{+}(φ_{l}α_{l})) ∈ Kerd^{+} ∩ B̄^{1}(M̂, ϵ_{l}), there is f_{l} ∈ B̄^{0}(M̂, ϵ_{l}) with φ_{l}α − df_{l} = Q(M̂)(d^{+}(φ_{l}α_{l})). Here |f_{l}|B̄^{0}(M̂, ϵ_{l}) are uniformly bounded and |Q(M̂)(d^{+}(φ_{l}α_{l}))|B̄^{1}(M̂, ϵ_{l}) → 0. Similarly we have g_{l} ∈ B̄^{0}(S^{3} × R, ϵ_{l}) with T_{2l}^{∗}((1 − φ_{l})α_{l}) − dg_{l} = Q(S^{3} × R)(T_{2l}^{∗}(d^{+}(1 − φ_{l})α_{l})).

Let us choose subindices k(l) >> l (with respect to {α_{l}}_{l}) and put ψ_{l} = φ_{k(l)}. Then we put β_{l} = ψ_{l}f_{l} + (1 − ψ_{l})((T_{2k(l)}^{−1})^{∗}g_{l}). Then we have I([α_{l}]) − [dβ_{l}] ∈ B̄_{0}^{1}({K_{2k(l)}, ϵ_{l}}_{l}). Since I : B̂^{∗}(M̂) ↦ B̂^{∗}(M̂) by [α_{l}] ∈ B̂^{∗}({K_{l}, ϵ_{l}}_{l}) → [φ_{l}α_{l} + (T_{2k(l)}^{−1})^{∗}((1 − φ_{l})α_{l})] ∈ B̂^{∗}({K_{2k(l)}, ϵ_{l}}_{l}) gives an isomorphism, this shows [α_{l}] = 0 ∈ H^{1}.

Now we verify the claim. Recall ∇_{i} = (d^{∗})_{w} ⊕d^{+} : (Ŵ_{1}^{2})_{w}(M̂, ϵ_{i}) ↦ L̂_{w}^{2}(M̂, ϵ_{i}) satisfies |∇_{i}| ≥ C_{i}. It is enough to see uniform bound C_{i} ≥ C. Suppose contrary. Let us take u_{i} ∈ (Ŵ_{1}^{2})_{w}(M̂, ϵ_{i}) with |∇_{i}(u_{i})|L̂_{w}^{2}(M̂, ϵ_{i}) → 0. Notice uniform bound:

|∇_{i}((1 − φ_{0})(u_{i}))|L̂_{w}^{2}(M̂, ϵ_{i}) ≥ C|(1 − φ_{0})(u_{i})|(Ŵ_{1}^{2})_{w}(M̂, ϵ_{i}). ∗

Then as in the proof of proposition 1.1, we have a familliar estimate:

|u_{i}|(Ŵ_{1}^{2})_{w}(M̂, ϵ_{i}) ≤ C{|φ_{0}u_{i}|L^{2+ϵ}(K_{0})_{0} + |∇_{i}(u_{i})|L̂_{w}^{2}(M̂, ϵ_{i})}.

Then using the compact embedding (W_{1}^{2+ϵ′})_{loc} ↪ L_{loc}^{2+ϵ}, ϵ′ ≤ ϵ, we find some u_{∞} ∈ Ker∇ ∩ (W_{1}^{2})_{w}. This must be zero. This shows |u_{i}|L^{2+ϵ}( Suppφ_{0}) → 0. Combining with the above ∗, one finds |u_{i}|L̂_{w}^{2}(M̂, ϵ_{i}) → 0. Since ∇_{i}(u_{i}) → 0, it follows from the elliptic estimate, |u_{i}|(Ŵ_{1}^{2})_{w}(M̂, ϵ_{i}) → 0. This completes the proof.

4 Asymptotic conformality

4.A Quasiconformal mappings: Let g and g′ be two Riemannian metrics on the same manifold M̂. Then the conformal distance is defined as:

d([g_{1}], [g_{2}])(x) = Sup_{ζ,ψ} log \{\frac{|ζ|_{2}}{|ψ|_{2}} : |ζ|_{1} = |ψ|_{1} = 1\}.

Let D, D′ ⊂ R^{4} be compact domains with Riemannian metrics g and g′ respectively. Then for a quasiconformal homeomorphism f : (D, g) ≅ (D′, g′), it has the following properties:

(1) f is almost everywhere differentiable with ∇f ∈ L_{loc}^{4}(D). The estimate holds:

∫_{D} |∇f|^{4} ≤ 16K^{3}vol(D′), K = ess supH(f).

(2) ([G]) The derivative ∇f is in fact in L_{loc}^{4+δ}, where δ > 0 is determined only by the quasiconformal constant, and ∇f is in the sense of ∫_{D} h\frac{∂f}{∂x_{i}} = − ∫_{D} f \frac{∂h}{∂x_{i}}.

(3) d([f^{∗}(g′)], [g]) is a bounded measurable function and for the quasiconformal constants H(f) in the introduction, the following pointwise estimate holds:

H(f)(x) = exp(d([f^{∗}(g′)], [g])_{x}).

(4) Let Λ^{2}(g) = Λ^{+}(g) ⊕ Λ^{−}(g) be the decomposition with respect to g. The projection to the antiself dual parts pr_{−}(g) : Λ^{2} ↦ Λ_{−}^{2}(g) depends only on the conformal class of g. Λ^{−}(f^{∗}(g′)) is determined by a bounded measurable bundle map μ = μ(f^{∗}(g′), g) : Λ^{−}(g) ↦ Λ^{+}(g) so that Λ^{−}(f^{∗}(g′)) can be expressed by the graph {(x, −μ(x)) : x ∈ Λ^{−}} = Λ^{−}(f^{∗}(g′)). The following pointwise estimate holds ([DS] p187):

\frac{1 + |μ|}{1 − |μ|}(x) ≤ H(f)(x)^{2} ≤ (\frac{1 + |μ|}{1 − |μ|}(x))^{2}.

(5) Let us put τ = τ(g, I^{∗}(g′)) = pr_{−}(g) − pr_{−}(f^{∗}(g′)). One may choose some canonical basis, {α_{±}^{i}}_{i} for Λ_{±}^{2}(g) (see [DS] p186). Then for α_{±} ∈ {α_{±}^{i}}_{i} ⊂ Λ^{±}(g), one has the equalities pr_{−}(α_{−}) = p(α_{−})^{−1}\{α_{−} + μ(α_{−})\}, p(a) = 1 − \frac{|μ(a)|^{2}}{|a|^{2}}, and −|x|^{2} +|μ(x)|^{2} −μ(x)α^{+} = 0, pr_{−}(α_{+}) = (x, μ(x)) where we use g norm. ¿From these inequalies, one gets the pointwise norm estimate:

\begin{aligned}|τ(g, f^{∗}(g′))| & = |pr_{−}(g) − pr_{−}(f^{∗}(g′))| \\ & ≤ (1 − |μ|^{2})^{−\frac{1}{2}}|μ| ≤ H(f)^{2} − 1. \end{aligned}

Lemma 4.1 (G, DS) Let f : (D, g) ↦ (D′, h) be a quasi conformal homeomorphism and p = 1, 2.

(1) Let α be a form in L^{\frac{4}{p}+ϵ}(Λ^{p}(D′)). Then f^{∗}(α) ∈ L^{\frac{4}{p}+ϵ′}(Λ^{p}(D)), where 0 ≤ ϵ′ = ϵ(p, δ, ϵ) < δϵ. Moreover there is C = C(ϵ, D, f, g, h) with C → [4 (ess supH(f))^{\frac{3}{2}}]^{\frac{p}{2}} as ϵ → 0 satisfying:

|f^{∗}(α)|L^{\frac{p}{4}+ϵ′}(D, g) ≤ C|α|L^{\frac{p}{4}+ϵ}(D, h).

(2) Let α be a function in W_{1}^{4}(D′, h). Then

∫ |f^{∗}(α)|^{4+ϵ′}vol_{g} ≤ C|α|^{4+ϵ′}W_{1}^{4}(D′, h)|∇f^{−1}|^{4}L^{4+δ}(D′, h).

P roof : For (1), p = 1, 2, one uses conformal invariance of L^{\frac{4}{p}} norms on p forms and ∇f ∈ L^{4+δ}. For (2), p = 0, one sees it as follows; recall the Sobolev embedding, L^{N} ↪ W_{1}^{4} for all N (see [GT] p167). Then by the Gehring’s theorem (2) above, one gets the estimate for a function α ∈ W_{1}^{4}:

∫ |f^{∗}(α)|^{4+ϵ′}vol_{g} ≤ ∫ |f^{∗}(α)|^{4+ϵ′}f^{∗}|∇f^{−1}|^{4}|∇f|^{4}vol_{g}

≤ C ∫ |f^{∗}(α)|^{4+ϵ′}f^{∗}|∇f^{−1}|^{4}f^{∗}(vol_{h}) ≤ C|α|^{4+ϵ′}L^{N}(D′, h)|∇f^{−1}|^{4}L^{4+δ}(D′, h)

≤ C|α|^{4+ϵ′}W_{1}^{4}(D′, h)|∇f^{−1}|^{4}L^{4+δ}(D′, h). [(\frac{N}{4 + ϵ′})^{−1} + (\frac{4 + δ}{4})^{−1} = 1)]

Thus one gets (2) above as desired.

Remark 4.1: Let φ : K ≅ K′ be a quasiconformal homeomorphism and D : K′ ≅ K′′ be a diffeomorphism. Let C and C′ be the quasiconformal constants of φ and D ∘ φ respectively. If ∇φ lies in L_{loc}^{4+ϵ}, then ∇(D ∘ φ) ∈ L_{loc}^{4+ϵ}, even C′ is larger than C.

Lemma 4.2 (see [GT] p 147) Let f : D ≅ D′ ⊂ R^{4} be a quasiconformal homeomorphism where D is compact. Let us take a smooth Riemannian metric g on D′. Then f^{∗}(g) can be approximated by a family of smooth Riemannian metrics g_{i} in the sense:

(1) g_{i} → f^{∗}(g) in L^{2}(D),

(2) quasiconformal constants K(g_{i}, f^{∗}(g)) are uniformly bounded, and

(3) H(g_{i}, f^{∗}(g))^{2} − 1, μ(f^{∗}(g), g_{i}), |pr_{−}(f^{∗}(g)) − pr_{−}(g_{i})| → 0 in all L^{n}.

P roof : We only consider (3). Let us consider H(g_{i}, f^{∗}(g))^{2} − 1. Its pointwise norms are uniformly bounded a.e.. Let us take local orthonomal basis e_{1}, …, e_{4} with respect to g_{i}. Then one may write f^{∗}(g) = Σg_{a,b}^{i}e_{a}^{∗} ⊗ e_{b}^{∗}, g_{a,b}^{i} − δ_{a,b} → 0 in L^{2}. Let us choose sections ζ and ξ with pointwise norms |ζ|_{g_{i}} = |ξ|_{g_{i}} = 1. Take small δ >> ϵ > 0. Then if we choose a sufficiently large i, then the measurable set D(δ) ⊂ D satisfying g_{a,b}^{i} −δ_{a,b} > δ on D(δ) has its measure less than ϵ. Thus we have the estimate (1 − δ)|ζ|_{g_{i}}^{2} ≤ |ζ|^{2} ≤ (1 + δ)|ζ|_{g_{i}}^{2} on D\D(δ). Combining with a uniform bound ess sup\frac{|ζ|}{|ξ|} ≤ K, we get the estimate:

\begin{aligned}|\frac{|ζ|^{2}}{|ξ|^{2}} − 1|^{2}L^{2}(D) & ≤ |\frac{|ζ|^{2}}{|ξ|^{2}} − 1|^{2}L^{2}(D\D(δ)) + (K^{2} − 1)^{2}ϵ \\ & ≤ (\frac{2δ}{1 − δ})vol(D) + (K^{2} − 1)^{2}ϵ. \end{aligned}

This shows H(g_{i}, f^{∗}(g))^{2} −1 → 0 in L^{2}. For μ and τ, we have similar inequality using the inequality in (5) above.

Next let us verify L^{n} convergence. By (2), ess sup K(g_{i}, f^{∗}(g)) are uniformly bounded. ¿From this and above, (3) will follow by the interpolation argument. Recall the intermediate inequality (see [GT] p146):

|u|L^{q} ≤ ϵ|u|L^{r} + ϵ^{−\frac{1−λ}{λ}}|u|L^{p}

where ϵ > 0, λ < 1, \frac{1}{q} = \frac{λ}{p} + \frac{(1−λ)}{r}, p ≤ q ≤ r. We show that if |u_{i}|L^{∞} ≤ C and |u_{i}|L^{2} → 0, then |u_{i}|L^{n} → 0 for all n. Take any small ϵ >> δ > 0. Then we choose a sufficiently large i so that |u_{i}|L^{2} < δ. We apply the above estimate as 1 < p ≤ 2, q = n, λ = \frac{1}{n}. With respect to u_{i}, we choose a sufficiently large r with |u_{i}|L^{r} ≤ 2|u_{i}|L^{∞} ≤ 2C. Then these data (q, r, λ) determines 1 < p ≤ 2. Now we have an estimate:

\begin{aligned}|u_{i}|L^{n} & ≤ ϵ|u_{i}|L^{r} + ϵ^{−\frac{1−λ}{λ}}|u|_{i}L^{p} \\ & ≤ 2Cϵ + Cϵ^{−\frac{1−λ}{λ}}|u_{i}|L^{2} ≤ 2Cϵ + Cϵ^{−\frac{1−λ}{λ}}δ. \end{aligned}

This implies (3). This completes the proof.

Remark 4.2: We donot know whether the above approximation in (3) could be done in L^{∞}. However picture of collapsing smooth structure may expect μ(f^{∗}(g), g_{i})|L^{N} approaches to zero non uniformly with respect to i and N.

4.B Asymptotic morphisms: Here we will introduce asymptotic equivalence relations between two asymptotically quasiconformally homeomorphisms. This generalizes the notion, asymptotic morphism introduced in the introduction.

Let M and N be smooth closed manifolds, and (M̂, g) and (N̂, g′) be the corresponding clylindrical ones. Suppose M̌ and Ň are equipped two ACH’s to (M̂, g) and (N̂, g′) with data respectively:

K_{0}(′) ⊂ ⋯ ⊂ M̌(Ň), I_{l}(′) : K_{l}(′) ↪ M̂(N̂).

We denote them as ACH_{1} : M̌ → (M̂, g) and ACH_{2} : Ň → (N̂, g′) respectively. We identify endM̂ and endN̂ with S^{3} × [0, ∞).

An asymptotic morphism from M̌ → (M̂, g) to Ň → (N̂, g′) consists of families of quasi conformal homeomorphisms:

E_{l} : K_{l} ≅ K′_{l}, F_{l} : I_{l}(K_{l}) ≅ I′_{l}(K′_{l})

satisfying:

(1) |H(F_{l})^{2} − 1|L_{loc}^{N}, |H(E_{l})^{2} − 1|L_{loc}^{N} → 0 l → ∞,

(2) Sup_{x,l}d(F_{l}(x), x) ≤ C, I_{l}(′)(x) ∈ I_{l}(K_{l}),

(3) d(F_{l′}(x), F_{l}(x)), d(E_{l′}(m), E_{l}(m)) → 0 as l, l′ → ∞ and

(4) |∇F_{l} − ∇F_{l′}|L_{loc}^{4+δ}(I_{l}(K_{l})), |∇E_{l} − ∇E_{l′}|L_{loc}^{4+δ}(K_{l}) → 0 for l′ ≥ l.

Let K_{0} ⊂ K_{1} ⊂ ⋯ ⊂ M̌ be an exaustion as before. Suppose M̌ is equipped with two asymptotic quasiconformal homeomorphisms to (M̂, g), ACH_{1} and to (N̂, g′), ACH_{2} with data:

K_{0} ⊂ ⋯ ⊂ M̌, I_{l}^{1} : K_{l} ↪ M̂, I_{l}^{2} : K_{l} ↪ N̂.

Then we say that these are self asymptotic equivalent, if {E_{l} = id } and {F_{l} = I_{l}^{2} ∘ (I_{l}^{1})^{−1}} consisit of an asymptotic morphism.

4.C Regularization: Let F_{i} : (D, g) ≅ (D′, g′) be a family of quasiconformal mapping with |H(F_{i})^{2} − 1|L_{loc}^{N} → 0 for a sufficiently large N >> 0. Recall τ_{i} = τ(g, F_{i}^{∗}(g′)) = pr_{−}(g) − pr_{−}(F_{i}^{∗}(g′)).

Let α be a smooth 1 form on D′ with d^{+}(α) = 0 and d(α) = w ≠ 0 for all points ([DS] p196). Let us take a constant δ > 0.

Lemma 4.3 Suppose |F_{i}^{∗}(w)|L_{loc}^{2+δ} are uniformly bounded. If F_{i} and ∇F_{i} converge in C_{loc}^{0} and L_{loc}^{4} respectively, then ∇F_{∞} ≡ lim ∇F_{i} in L_{loc}^{4+δ′} for some 0 < δ′ < δ.

P roof : By the condition, there are measurable sets D_{i} ⊂ D satisfying m(D_{i}) → 0 and (2) ess supτ_{i}|D\D_{i} < c. Then there is C with a pointwise bound |∇F_{i}|^{4+δ′} ≤ C|F_{i}^{∗}(w)|^{2+δ}. Thus ∇F_{i}|D\D_{i} ∈ L_{loc}^{4+δ′}. Then we show limit ∇F_{i} ≡ ∇F_{∞} ∈ L_{loc}^{4+δ′}, δ′ = δ′(c). Let us choose small ϵ > 0. Then one takes a subsequence k(l) with m(D(ϵ)) < ϵ, D(ϵ) = ∪_{l}D_{k(l)} and ∇F_{k(l)}|D\D(ϵ) ∈ L_{loc}^{4+δ′}. Recall the inequality |u|L^{q} ≤ ϵ|u|L^{r} + ϵ^{−\frac{1−λ}{λ}}|u|L^{p} where q^{−1} = λ/p + (1 − λ)/r, p ≤ q ≤ r. Since |∇F_{k(l)}|L_{loc}^{4+δ′}(D\D(ϵ)) are uniformly bounded, one may apply this inequality to u = ∇F_{k(l)} − ∇F_{k(l′)}. Then one sees that for all ϵ > 0 and some 0 < δ′′ < δ′, |∇F_{∞}|L_{loc}^{4+δ′′}(D\D(ϵ)) are uniformly bounded. We change the notation δ′′ by δ′. Then by letting ϵ → 0, one knows ∇F_{∞} ∈ L_{loc}^{4+δ′}. By changing indices, one may assume |∇F_{∞} − ∇F_{i}|L_{loc}^{4+δ′}(D\D_{i}) → 0. This completes the proof.

Now we consider the converse direction.

Lemma 4.4 Suppose all F_{i} above converges in L_{loc}^{4+δ}. Then there are constants C, 0 ≤ ϵ′ < ϵ such that for any u ∈ B̄^{∗}(D′, ϵ)_{loc}, there is a bound:

|F_{∞}^{∗}(u)|B̄^{∗}(D, ϵ′)_{loc} ≤ C|u|B̄^{∗}(D′, ϵ)_{loc}.

P roof : As above there are measurable sets D_{i} ⊂ D satisfying (1) m(D_{i}) → 0 and (2) ess supτ_{i}|D\D_{i} < c. Then there is C with a pointwise bound |∇F_{i}|^{4} ≤ C det (∇F_{i}) on D\D_{i}. Then as [DS] p198, one get the estimate:

|F_{i}^{∗}(u)|B̄^{∗}(D\D_{i}, ϵ′) ≤ C|u|B̄^{∗}(D′\D′_{i}, ϵ).

Now let us choose a subindices k(l) satisfying lim_{l}m(∪_{l′≥l}D_{k(l′)}) = 0. We put E_{l} = ∪_{l′≥l}D_{k(l′)} ⊂ D. Then for all triple l′′ ≥ l′ ≥ l, there is a uniform estimate:

|(F_{k(l′′)}−F_{k(l′)})^{∗}(u)|B̄^{∗}(D\E_{l}, ϵ′) ≤ C(|∇(F_{k(l′′)}−F_{k(l′)})|L^{4+δ}(D))|u|B̄^{∗}(D′\E′_{l}, ϵ).

This shows |F_{∞}^{∗}(u)|B̄^{∗}(D\E_{l}, ϵ′) ≤ C|u|B̄^{∗}(D′\E′_{l}, ϵ). Since m(E_{l}) → 0, this gives the result.

4.D Induced morphisms: Let us take two closed smooth four manifolds M and M′. Suppose (M̂, g) and (M̂′, g′) admit a uniformly bounded asymptotic morphism, consisting of families of quasiconformal homeomorphisms {F_{l} : (K_{l}, g) ≅ (K′_{l}, g′)}_{l}. Recall τ_{l} = τ(F_{l}^{∗}(g′), g) = pr_{−}(g) − pr_{−}(F_{l}^{∗}(g′)) → 0 in L^{n} for a sufficiently large n.

Let B̂^{∗}(M̂(′)) = B̂^{∗}({K_{l}(′)}_{l}, ϵ(′)) and (B̂^{i}(M̂(′)), d_{i}) be the AHS complex:

0 ⟶ B̂^{0}(M̂(′)) ⟶^{d} B̂^{1}(M̂(′)) ⟶^{d^{+}} B̂^{2}(M̂(′)) ⟶ 0.  (4.1)

Proposition 4.1 Choosing sufficiently small constants ϵ′ < ϵ and δ′ < δ, there is an induced morphism:

F̄^{∗} : (B̂^{i}(M̂, w, ϵ), d_{i}) ↦ (B̂^{i}(M̂′, w′, ϵ′), d_{i}).

Sublemma 4.1 Let F : (K, g) ≅ (K, h) be a quasiconformal homeomorphism between two Riemannian spaces with τ(F^{∗}(h), g)|L^{n} < α for some large n. Then there is c = c(α, F) > 0 and ϵ′ < ϵ with:

|[F^{∗}, d^{+}]u|L^{2+ϵ′}(K, g) ≤ c|du|L^{2+ϵ}(K, h), c → 0 as α → 0.

Proof: Let d|Λ^{2} = d^{+} ⊕ d^{−} be the decomposition with respect to g, and express d^{+}(F^{∗}(h)) = d^{+} + τd. Then one has the equality:

[F^{∗}, d^{+}] = F^{∗} ∘ d^{+}(h) − d^{+}(g) ∘ F^{∗} = F^{∗} ∘ (1 + ∗(h))d − d^{+}(g) ∘ F^{∗}
= (1 + F^{∗} ∗ (h)(F^{−1})^{∗})d ∘ F^{∗} − d^{+}(g) ∘ F^{∗}
= (d^{+}(F^{∗}(h)) − d^{+}(g)) ∘ F^{∗} = τd ∘ F^{∗}.

Then the following estimates completes the proof:

\begin{aligned}|[F^{∗} & , d^{+}]u|L^{2+ϵ′}(K_{l}, g) = |τdF^{∗}u|L^{2+ϵ′} \\ & ≤ |τ|L^{n}|dF^{∗}u|L^{2+ϵ′′} ≤ C(|∇F |L^{4+δ})|τ|L^{n}|du|L^{2+ϵ}(K_{l}, h). \end{aligned}

Proof of proposition: Here we claim the following ; let f and g be functions over M̂ with a uniform bound |f|L_{loc}^{2+ϵ′} ≤ C|g|L_{loc}^{2+ϵ}, ϵ′ < ϵ. Then for some choice of δ′ < δ, one gets the global bound:

|f|L_{w′}^{2+ϵ′} ≤ C|g|L_{w}^{2+ϵ}.

Proof of claim: We have estimates:

∫ |f|^{2+ϵ′}exp(w′) ≤ Σ_{n}|g_{i}|^{2+ϵ′}L_{loc}^{2+ϵ}exp(δ′n)

= Σ_{n}(|g_{i}|L_{loc}^{2+ϵ}exp(\frac{δ′′n}{2 + ϵ′}))^{2+ϵ′}(exp(δ′ − δ′′)n)

≤ (Σ_{n}|g_{i}|^{2+ϵ}L_{loc}^{2+ϵ}exp(δn))^{p^{−1}}(Σ_{n}exp(qn(δ′ − δ′′))^{q^{−1}} ≤ C|g|^{2+ϵ′}L_{w}^{2+ϵ}

where p = \frac{2+ϵ}{2+ϵ′}, p^{−1} + q^{−1} = 1 and δ′′ = p^{−1}δ > δ′. This verifies the claim.

Proof of proposition: Let us take [u_{l}] ∈ B̂^{∗}(M̂). Then by the estimate in lemma 4.1, lemma 4.4 and the above claim, one gets:

||F_{l}^{∗}(u_{l})||B̄^{∗}(K′_{l}, g′, w′, ϵ′) ≤ C||u_{l}||B̄^{∗}(K_{l}, g, w, ϵ).

(More precisely one should remove some small measurable sets, or use F_{∞}^{∗}. However as far as considering asymptotic sequences, these do not matter and we will denote as above.)

Let us check [F_{l}^{∗}(u_{l})] defines also an element in B̂^{∗}(M̂′). Recall |∇(F_{l} − F_{l′})|L^{4+δ} → 0. Then we have the inequalities:

\begin{aligned}||F_{l} & ^{∗}(u_{l}) − F_{l′}^{∗}(u_{l′})||B̄^{∗}(K′_{l}, g′, ϵ′) = ||F_{l′}^{∗}[u_{l} − u_{l′}]|| + ||(F_{l} − F_{l′})^{∗}(u_{l})|| \\ & ≤ C(F_{l}, ϵ)||u_{l} − u_{l′}||B̄^{∗}(K_{l}, g, ϵ) + δ(l, l′)||u_{l}|| → 0. \end{aligned}

Thus F^{∗} : B̂^{∗}(M̂, ϵ) ↦ B̂^{∗}(M̂′, ϵ′) gives a bounded map.

Next let us consider the differentials. Notice that d : B̂^{0}(M̂) ↦ B̂^{1}(M̂) commutes with F^{∗}.

Next let us consider d^{+} and the family F_{l}^{∗}(d^{+}(u_{l})) ∈ L̂_{w′}^{2}(K′_{l})_{0}. By lemma 4.4 and and the above claim, the family is uniformly bounded. Then we show ||d^{+}F_{l′}^{∗}u_{l′} − d^{+}F_{l}^{∗}u_{l}||B̄^{2}(K_{l′}, g_{l′}, w′, ϵ′) → 0. In fact one has the estimates:

||d^{+}F_{l′}^{∗}u_{l′} − d^{+}F_{l}^{∗}u_{l}||B̄^{2}(K_{l′}, g_{l′}, w′, ϵ′)
≤ ||[F_{l′}^{∗}, d^{+}]u_{l′}|| + ||[F_{l}^{∗}, d^{+}]u_{l}|| + ||F_{l′}^{∗}d^{+}(u_{l′}) − F_{l}^{∗}d^{+}(u_{l})||

≤ δ(F_{l}, F_{l′}, ϵ){||du_{l′}||L̂_{w}^{2} + ||du_{l}||L̂_{w}^{2}} + C||du_{l} − du_{l′}|| + Cδ(l, l′)||du_{l}||.

By sublemma 4.1 and |τ_{l}|L^{n} → 0, one may assume that the last term converges to zero. Thus F^{∗} gives a morphism from {B̂^{∗}(M̂, ϵ), d^{∗}} to {B̂^{∗}(M̂′, ϵ′), d^{∗}}. This completes the proof.

Corollary 4.1 Let M̂ and M̂′ be asymptotically q.c. equivalent. Let us take ϵ > ϵ′ > ϵ′′ > 0. Then we have the induced maps:

F^{∗} : B̂^{∗}(M̂, w, ϵ) ↦ B̂^{∗}(M̂′, w′, ϵ′), (F^{−1})^{∗} : B̂^{∗}(M̂′, w′, ϵ′) ↦ B̂^{∗}(M̂, w′′, ϵ′′)

such that the composition (F^{−1})^{∗} ∘ F^{∗} gives the identity.

5 Connections and bundles

Let a cylindrical manifold (M̂, g) be as before. For all the later sections, we will use the underlying Banach spaces B̂^{∗}(M̂, ϵ) defined in section 2.A (3.C), where all the constants ϵ_{l} are the same ϵ > 0.

Let E → M̂ be a G bundle (G = SU(2) or SO(3)), and fix a trivialization of E on the end. This determines c_{2} or p_{1} of E. Let T ⊂ R_{+} be an interval. Then for g ∈ G, we put T (g) ∈ Aut(S^{3}×T ×G) as the constant gauge transformation. We regard S^{3} × T ⊂ S^{3} × R_{+} = endM̂.

Let us fix a smooth connection A_{0} which is trivial on the end. Then one defines a formal affine Banach space and a Banach Lie group:

\begin{matrix}As𝔄 = \{A_{0} + a : a ∈ B̂^{1}\}, \\ As𝔊 = \{u = [u_{l}] : [∇_{A_{0}}u_{l}] ∈ B̂^{1}, u_{l} ∈ B̄_{loc}^{0}(Aut(E)), ∗\} \end{matrix}

where ∗ is as follows; recall we have families of indices b(l, j), j = 0, 1, …, n. Let us take a constant c = c(u) > 0 and put a family of intervals

\begin{matrix}M(l, 0) = S^{3} × [c, b(l, 0) − c], \\ M(l, j) = S^{3} × [b(l, j) + c, b(l, j + 1) − c] ⊂ endM̂ \end{matrix}

where j ≥ 1 and b(l, n + 1) = ∞. Then for some g = {g_{0}(u), …, g_{n}(u)} ⊂ G and for a locally constant gauge transformation T (g)_{l} = Σ_{j=0}^{n}M(l, j) × g_{j}, one has [u_{l} − T (g)_{l}] ∈ B̂^{0}.

Naturally one has stratifications:

As𝔄_{0} ⊂ As𝔄_{1} ⊂ …, As𝔊_{0} ⊂ As𝔊_{1} ⊂ …

where As𝔄_{0} = {A_{0} + a : a ∈ B̄^{1}(M̂)}, As𝔊_{0} = {u ∈ As𝔊 : ∇_{A_{0}}u ∈ B̄^{0}(M̂)}. It is straightforward to check the following:

Lemma 5.1 (1) As𝔊 is a Banach Lie group. Its Lie algebra As𝔤 = Lie As𝔊 = {σ ∈ B̂_{loc}^{0} : ∇_{A_{0}}σ ∈ B̂^{1}, ∗} with norm ||∇_{A_{0}}σ||B̂^{1} + Σ_{j=0}^{n}|h_{j}(σ)|, h_{j}(σ) ∈ 𝔤.

(2) As𝔊 acts on As𝔄 by:

[u_{l}](A_{0} + [a_{l}]) = [u_{l}^{∗}(A_{0} + a_{l})].

P roof : (1) Recall the local Sobolev embedding C^{0} ↪ W_{1}^{4+2ϵ}. ¿From this, we get the multiplication map:

B̂^{0}(M̂; ϵ) × B̂^{0}(M̂; ϵ) ↦ B̂^{0}(M̂; ϵ).

By the Sobolev embedding, any element g ∈ B̂^{0}(M̂; ϵ) consists of a family of continuous sections g_{l} : K_{l} ⊂ M̂ ↦ Aut(E). Thus one can define exponential map, and passing through it, one gets a manifold structure on As𝔊.

(2) follows from the Sobolev multiplication properties:

B̂^{0} × L̂_{w}^{p} ↦ L̂_{w}^{p}, L̂_{2w}^{4+2ϵ} × L̂_{2w}^{4+2ϵ} ↦ L̂_{w}^{2+ϵ}

where p = 4 + 2ϵ or 2 + ϵ. Then we check the condition:

||u_{l}^{∗}(A_{0} + a_{l}) − u_{l′}^{∗}(A_{0} + a_{l′})||B̄^{1}
≤ C[(1 + ||u_{l}||B̄^{0} + ||u_{l′}||B̄^{0})^{2}||u_{l} − u_{l′}||B̄^{0}]
+ ||(u_{l} − u_{l′})^{∗}a_{l}||B̄^{1} + ||u_{l′}^{∗}(a_{l} − a_{l′})||B̄^{1}
≤ C(1 + ||u_{l}||B̄^{0} + ||u_{l′}||B̄^{0})^{2}
[||u_{l} − u_{l′}||B̄^{0} + ||u_{l} − u_{l′}||B̄^{0}||a_{l}||B̄^{1} + ||a_{l} − a_{l′}||B̄^{1}] → 0.

This completes the proof.

Let us consider a smooth map and its kernel:

\begin{matrix}F^{+} : As𝔄 ↦ B̂^{2}(M̂; Λ_{+}^{2} ⊗ ad𝔤), [A_{0} + a_{l}] → [F_{A_{0}+a_{l}}^{+}]. \\ As𝔐̂ = \{[A_{l}] ∈ As𝔄|[F_{A_{l}}^{+}] ∈ B̄_{0}^{2}(\{K_{l}\}_{l}, ϵ) = 0 ∈ B̂^{2}(\{K_{l}\}_{l}, ϵ)\}. \end{matrix}

We call an element [A_{l}] ∈ As𝔐̂(M̂, ϵ) an asymptotic ASD connection. As𝔊, or As𝔊_{k}, 0 ≤ k ≤ n, also act on As𝔐̂ or As𝔐̂_{k}, and one puts:

\begin{matrix}As𝔐 ≡ As𝔐̂/As𝔊, As𝔐_{k} ≡ As𝔐̂_{k}/As𝔊_{k}, \\ 𝔐(M̂) = As𝔐_{0} ⊂ As𝔐_{1} ⊂ ⋯ ⊂ As𝔐_{n} = As𝔐. \end{matrix}

Recall that in order to get the underlying Banach spaces, one has fixed pair of indices and translations a = {a(l, j)} and T = {T(l, j)} satisfying some propeorties in 2.A. Then we get the parametrized asymptotic ASD moduli space:

P As𝔐_{k}(M̂, g, ϵ) = ∪_{(a,T)}As𝔐_{k}(M̂, g, ϵ; a, T).

When we consider a single asymptotic moduli space, then we will not specify about (a, T).

5.B Regularity: The following lemma is well known.

Lemma 5.2 Let us take any [A] ∈ 𝔐(M̂, g). Then there are positive δ_{k} > 0 such that one may represent it by a smooth ASD connection A with:

|A|endM̂|C^{k}(S^{3}, t) ≤ C_{k}exp(−δ_{k}t)

identifying endM̂ ≅ S^{3} × [0, ∞).

Let φ_{l} be the family of cut off functions over M̂ in 2.A. Then one can assign I([A]) = [A_{l} = A_{0} + φ_{l}a], A = A_{0} + a which defines an element in As𝔄(M̂). It follows from the above exponential decay of A, that [A_{l}] defines an element in As𝔐̂. Thus one gets a map Î: 𝔐̂′ ↦ As𝔐̂, where 𝔐̂′ is choice of lift as above for every element in 𝔐.

Let us take [A_{l}] ∈ As𝔐̂.

Lemma 5.3 (1) Any asymptotic ASD connection [A_{l}] converges as l → ∞ to (A(0), …, A(n)), B̄^{1}(M̂, ϵ) and B̄^{1}(S^{3} × R, ϵ) ASD connections respectively.

(2) Moreover if [A_{l}] and [A′_{l}] are As𝔊(M̂, ϵ) equivalent, and if they correspond to A(j) and A(j)′ respectively, then each A(j) and A(j)′ are 𝔊(M̂, ϵ) or 𝔊(S^{3} × R, ϵ) equivalent.

(3) There exists [g_{l}] ∈ As𝔊 so that [g_{l}][A_{l}] ∈ As𝔄 can be represented by another [A′_{l}] where each A′_{l} are given by cutting the same family of smooth ASD connections (A(0), …, A(n)).

P roof : We only consider n = 1, one bubble case. The general case is similar. Let us take [A_{l}] ∈ As𝔐 and consider the family of B̄^{1} connections A_{l} over M̂. By definition, ||F_{A_{l}}^{+}||L̂_{w}^{2+ϵ}(M̂) → 0 as l → ∞, and A_{l} converges to A(0) and A(1) over M̂ and S^{3} × R respectively in B̄^{1} topology. Each A(i) satisfies the ASD equation with finite L^{2} curvature norms. If for some [g_{l}] ∈ As𝔊, ||g_{l}^{∗}A_{l} − A′_{l}||B̄^{1} → 0, then we get g_{l} → (g(0), g(1)) ∈ B̄^{0} × B̄^{0}, and clearly this satisfies g(i)^{∗}A(i) = A(i)′. This verifies (2).

Let A(i) be as above. By the above lemma, there exist gauge transformations g(0) and g(1) over M̂ and S^{3} × R such that g(i)^{∗}(A(i)) are smooth ASD connections. We check g(i) ∈ 𝔊 ⊂ As𝔊. On the end one can put A(i) = d + a_{i} and g(i)^{∗}(A(i)) = d + a′_{i} where a_{i}, a′_{i} ∈ B̄^{1}. Then one has the estimate:

\begin{matrix}|a_{i}− & a′_{i}|L̂_{w}^{4} = |g(i)^{∗}(A(i)) − A(i)|L̂_{w}^{4} = |g(i)^{−1}dg(i) + g(i)^{−1}a_{i}g(i) − a_{i}|L̂_{w}^{4} \\ ≥ |dg(i)|L̂_{w}^{4} − 2|a_{i}|L̂_{w}^{4}. \end{matrix}

This shows dg(i) ∈ L̂_{w}^{4} and subtracting some constants, one gets g(i) ∈ B̄^{0}.

Let S_{l} be the translations in lemma 2.3. We put A′_{l} = d+φ_{l}a′_{1} +(1−φ_{l})S_{2l}^{∗}a′_{2}. Since g(i) ∈ B̄^{0}, there are g′_{i} ∈ G so that |g(0) − g′_{0}|C^{0}(S^{3} × {T }), |g(1) − g′_{1}|C^{0}(S^{3} × {−T }) → 0 as T → ∞. After multiplying some constant, one may assume g′ = g′_{0} = g′_{1} ∈ G. Again using cut off functions, like h_{l} = g′ + φ_{l}(g(0) − g′) + (1 − φ_{l})S_{2l}^{∗}(g(1) − g′), one gets an element [h_{l}] ∈ As𝔊(M̂) satisfying [h_{l}][A_{l}] = [A′_{l}]. This verifies (3). This completes the proof.

Corollary 5.1 For any pair ϵ > ϵ′ > 0, the natural inclusions give homeomorphisms:

I : As𝔐_{k}(M̂, g, ϵ) ≅ As𝔐_{k}(M̂, g, ϵ′) 0 ≤ k ≤ n.

Corollary 5.2 There exists a surjection (essentially an isomorphism):

As𝔐(M̂; E) ↦ ∪_{{E_{l}}}𝔐(M̂; E_{0}) × ×_{j=1}^{n}𝔐(S^{3} × R; E_{j})

where (1) w_{2}(E) = w_{2}(E_{0}) and (2) p_{1}(E) = Σ_{j=0}^{n}p_{1}(E_{j}).

In particular there is a natural injection I : 𝔐(M̂, g′) ↪ As𝔐(M̂) which is induced by Î above.

5.C Local structure of the moduli spaces: Let us consider the action:

As𝔊 × As𝔄 ↦ As𝔄.

The derivative at (id, A) is given by:

(u, a) ↦ a + d_{A}(u) ∈ B̂^{1}.

Lemma 5.4 For any A ∈ As𝔐̂(M̂), {B̂^{∗}, d_{A}^{∗}} is a Fredholm complex with the index −2p_{1}(E) − 3(1 + b_{+}^{2}(M)). We denote their cohomology groups by H_{A}^{∗} = Kerd_{A}^{∗+1}/ imd_{A}^{∗}.

P roof : Let us take A = [A_{l}] ∈ As𝔐̂(M̂). First let us see the composition d_{A}^{+} ∘ d_{A}^{0} = 0. Since d_{A}^{+}d_{A}([u_{l}]) = [F_{A_{l}}^{+}, u_{l}], it follows from the Sobolev multiplication B̂^{2} × B̂^{0} ↦ B̂^{2}.

Let us see imd_{A}^{0} is closed. By 5.B, one may choose [A′_{l}], As𝔊 gauge equivalent to [A_{l}] such that A_{l} = φ_{l}A_{0} + (1 − φ_{l})S_{2l}^{∗}A_{1}, where A_{j} are smooth ASD connections over M̂ and S^{3} × R respectively. Then for each l, d_{A′_{l}} : B̄^{0}(M̂; ad𝔤) ↦ B̄^{1}(M̂; Λ^{1}⊗ad𝔤) has closed range with finite dimensional kernel (c.f. [K] lemma 1.5). Then as the proofs of proposition 1.1 and lemma 3.2, one sees that d_{A}^{0} : B̂^{0} ↦ B̂^{1} has closed range. Similarly one can check d_{A}^{+} : B̂^{1} ↦ B̂^{2} has closed range. The index computation is done from corollary 2.1, corollary 5.2 and the sum formula. This completes the proof.

Let us take A ∈ As𝔐. Then there exists Q_{A} : B̂^{2} ↦ B̂^{1} as in the last part of the proof of lemma 3.2. We recall its construction below. First we will construct Q_{A} : imd_{A}^{+} ∩ B̄^{2}(M̂) ↦ B̄^{1}(M̂) with d_{A}^{+} ∘ Q_{A} = id for an ASD connection A over M̂. Similarly we have Q_{A} over S^{3} × R. Then let A ∈ As𝔐 correspond to a family of ASD connections (A_{0}, …, A_{n}). As in the last part of the proof of lemma 3.2, we cut and connect Q_{A_{j}}’s as:

Q′_{A} = Σψ_{j}Q_{A_{j}}ψ′_{j} : imd_{A}^{+} ∩ B̂^{2}({K_{l}}_{l}; ϵ) ↦ B̂^{1}({K_{l}}_{l}; ϵ).

Again as in the lemma 3.2, one can modify Q′_{A} to Q_{A} so that Q_{A} satisfies d_{A}^{+} ∘ Q_{A} = id. This induces the desired map Q_{A} : B̂^{2} ↦ B̂^{1}.

Now let us make the first construction. Let us take the bounded operator:

Q_{A} : imd_{A}^{+} ∩ L_{w}^{2}(M̂; Λ_{+}^{2} ⊗ ad𝔤) ↦ Ker(d_{A})_{w}^{∗} ⊂ (W_{1}^{2})_{w}(M̂; Λ^{1} ⊗ ad𝔤)

with d_{A}^{+} ∘ Q_{A} = id. We claim that if u ∈ L^{2+ϵ} ∩ L_{w}^{2}, then Q_{A}(u) ∈ L_{w}^{2+ϵ} and the norm of Q_{A}(u) is uniformly bounded by the one of u. Recall the Sobolev inequality | |L_{loc}^{2+ϵ} ≤ C| |(W_{1}^{2})_{loc} where C is independent of choice of small ϵ. Let {φ_{l}}_{l} be the exausting family of cut off functions on M̂. Thus φ_{l}Q_{A}(u) ∈ L_{w}^{2+ϵ}. Then we use the Sobolev inequality, the elliptic estimate and the local to global method used in the proof of sublemma 3.2. Thus one gets the following estimate:

\begin{aligned}|φ_{l} & Q_{A}(u)|L_{w}^{2+ϵ}(M̂) ≤ C|φ_{l}Q_{A}(u)|L_{w}^{2}(M̂) + C|d_{A}^{+}(φ_{l}Q_{A}(u))|L_{w}^{2} \\ & ≤ C|φ_{l}Q_{A}(u)|L_{w}^{2}(M̂) + C|φ_{l}u|L_{w}^{2}(M̂) + C|Q_{A}(u))|L_{w}^{2}( Suppdφ_{l}) \end{aligned}

where the last term is uniformly bounded. Moreover |(φ_{l}−φ_{l′})Q_{A}(u)|L_{w}^{2+ϵ}(M̂) → 0 as l, l′ → ∞. One may choose all the constants uniformly bounded independently of ϵ.

Again by the elliptic estimate, one gets Q_{A} : imd_{A}^{+} ∩ L̂_{w}^{2} ↦ (Ŵ_{1}^{2})_{w}. Then by the estimate of the last part of section 1, one gets the desired extension Q_{A} : imd_{A}^{+} ∩ B̄^{2} ↦ B̄^{1}. This finishes the construction of Q_{A} : imd_{A}^{+} ∩ B̂^{2} ↦ B̂^{1}.

Thus B̂^{1} splits as:

B̂^{1} = imQ_{A} ⊕ kerd_{A}^{+}

by u = u′ + u − u′, u′ = Q_{A}d_{A}^{+}(u). ¿From this, one has a natural isomorphism:

kerd_{A}^{+}/ imd_{A} ≅ B̂^{1}/ imQ_{A} + imd_{A}.

In particular imQ_{A}+ imd_{A} ⊂ B̂^{1} has finite codimension. By adding a finite dimensional subspace V_{A} ⊂ B̂^{2}, one has a tangent space of As𝔅 = As𝔄/As𝔊 at [A]:

T_{[A]}As𝔅 ≅ imQ_{A} ⊕ V_{A}.

Let us put:

\begin{matrix}CG(A) = \{A + a|a ∈ imQ_{A} + V_{A}\}, \\ I : CG(A) × As𝔊 ↦ As𝔄 (A′, g) → g^{∗}(A′). \end{matrix}

Then dI is given by:

dI = id ⊕ d_{A} : ( imQ_{A} ⊕ V_{A}) × As𝔤 = LieAs𝔊 ↦ B̂^{1}.

Thus if A is irreducible (Kerd_{A} = 0), then one gets the following:

Proposition 5.1 There are neighbourhoods U ⊂ As𝔄 of A and V ⊂ As𝔊 of id such that:

\begin{matrix}I : U ∩ CG(A) × V ↦ As𝔄, \\ I : U ∩ CG(A) ∩ As𝔐̂ × V ↦ As𝔐̂ \end{matrix}

are homeomorphisms respectively.

In particular if d_{A}^{+} is surjective, then at [A], As𝔊 acts freely on As𝔐 and As𝔐_{k}\As𝔐_{k−1}. As𝔐/As𝔊 and As𝔐_{k}\As𝔐_{k−1}/As𝔊 are C^{∞} manifolds of dimension −2p_{1}(E) − 3(1 + b_{+}^{2}(M)).

Here is one remark. Suppose one is given a uniformly bounded asymptotic equivalence {I_{l}}_{l} between (M̂, g) and (M̂′, g′). Let us take A = [A_{l}] ∈ As𝔐(M̂). Let us choose small constants ϵ > ϵ′ > ϵ′′. Then one gets the following commutative diagram:

\begin{aligned}d_{A}^{+} & B̂^{1}(M̂, ϵ) & ↦ & B̂^{2}(M̂, ϵ) \\ & I^{∗} ↓ & & I^{∗} ↓ \\ d_{I^{∗}(A)}^{+} & B̂^{1}(M̂′, ϵ′) & ↦ & B̂^{2}(M̂′, ϵ′) \\ & (I^{−1})^{∗} ↓ & & (I^{−1})^{∗} ↓ \\ d_{A}^{+} & B̂^{1}(M̂, ϵ′′) & ↦ & B̂^{2}(M̂, ϵ′′). \end{aligned}  (5.1)

¿From this diagram, it follows if d_{I^{∗}(A)}^{+} is surjective, then for any element u ∈ B̂^{2}(M̂, ϵ), there is α ∈ B̂^{2}(M̂, ϵ′′) with d_{A}^{+}(α) = u.

6 Transversality

Let (M̂, g) be a smooth cylindrical manifold, and E → M̂ be an SO(3) bundle with a fixed trivialization on the end. Thus the bundle determines w_{2}(E) ∈ H^{2}(M : Z_{2}) and p_{1}(E) ∈ H^{4}(M : Z). Here we quickly review the construction of (W_{l}^{2})_{w} ASD moduli space over (E, M̂, g).

One has already used an weight function w : M̂ ↦ [0, ∞). Let us choose a smooth connection A_{0} over E with A_{0}|end = d. Then let us put an affine Hilbert space and a Hilbert Lie group (l ≥ 3):

\begin{matrix}𝔄 = \{A_{0} + a : a ∈ (W_{l}^{2})_{w}(M̂; ad𝔤 ⊗ Λ^{1})\}, \\ 𝔊 = \{g ∈ C_{loc}^{0}(AutE) : ∇_{A_{0}}g ∈ (W_{l}^{2})_{w}(M : ad𝔤 ⊗ Λ^{1})\}. \end{matrix}

Moreover put 𝔐̂ = {A ∈ 𝔄 : F_{A}^{+} = 0}. 𝔊 acts on both 𝔄 and 𝔐̂, and one denotes their quotient spaces by 𝔅 = 𝔄/𝔊 and 𝔐 = 𝔐̂/𝔊 respectively. Sometimes one denotes 𝔐 = 𝔐(g).

Lemma 6.1 Let A ∈ 𝔐̂. Then the AHS complex:

\begin{aligned}0 ⟶ (W_{l+1}^{2})_{w}(M̂, g; ad𝔤) & ⟶^{d_{A}} (W_{l}^{2})_{w}(M̂, g; ad𝔤 ⊗ Λ^{1}) \\ & ⟶^{d_{A}^{+}} (W_{l−1}^{2})_{w}(M̂, g; ad𝔤 ⊗ Λ_{+}^{2}) ⟶ 0. \end{aligned}

is of Fredholm with the index −2p_{1}(E) − 3b_{+}^{2}(M).

This follows from the usual excision method.

6.B Transversality for (W_{k}^{2})_{w} spaces: The aim in 6.B is to see:

Proposition 6.1 Suppose b_{+}^{2}(M) ≥ 1. Then for some generic choice of g′, 𝔐(g′) is a finite dimensional smooth manifold of dimension −2p_{1}(E) − 3(1 + b_{+}^{2}(M)) (note M is simply connected).

In order to construct smooth moduli spaces, one uses K.Uhlenbeck’s generic metric theorem. The proof is parallel to [FU] as far as one uses Hilbert spaces. Later one will see that this perturbation also can be applicable to L_{w}^{p} spaces. Let (M̂, g, w) be a cylindrical manifold, and choose a proper map h : M̂ ↦ [0, ∞) with h(x) ≥ \frac{w}{2}(x). Then one introduces the following Banach manifold:

ℭ = {ϕ ∈ C^{l}(Gl(T M̂)) : limsup_{i}(Σ_{j=0}^{l}e^{h}|∇^{j}(ϕ^{∗}g − g)|S^{3} × [i, i + 1] = 0}.

Let us take ϕ ∈ ℭ, and put g′ = ϕ^{∗}g. Then one has the AHS complex with respect to g′:

0 ⟶ (W_{l+1}^{2})_{w}(M̂, g′) ⟶^{d} (W_{l}^{2})_{w}(M̂, g′) ⟶^{d^{+}} (W_{l−1}^{2})_{w}(M̂, g′) ⟶ 0.

One sees that this is a Fredholm complex with the same index as the unpurturbed one.

Lemma 6.2 Suppose M is indefinite. Then by a small perturbation of the Riemannian metric, there are no orbit of reducible connections in 𝔐_{k}(M̂).

In order to verify this, we follows [FU]. Let us take any compactly supported smooth connection A_{0} over M̂, and consider 𝔄(M̂) = {A_{0}+a|a ∈ (W_{l}^{2})_{w}(M̂, g)}. Then one introduces a 𝔊(M̂) equivariant map:

P_{+} : 𝔄(M̂) × ℭ ↦ (W_{l−1}^{2})_{w}(M̂, g)

by P_{+}(A, ϕ) = P_{+}(g)(ϕ^{−1})^{∗}F_{A}, where P_{+}(g) is the projection to the self dual part with respect to g. Let us put 𝔐̄(M̂) = P_{+}^{−1}(0).

Proposition 6.2 (FU) 𝔐̄(M̂) ∩ 𝔄^{∗}(M̂) × ℭ/𝔊(M̂) is a smooth Banach manifold where 𝔄^{∗} consists of irreducible connections.

We sketch its proof. First we see that dP_{+} is surjective at any (A, φ) with P_{+}(A, φ) = 0. Then it follows that 𝔐̄(M̂) is a Banach manifold on which 𝔊(M̂) acts. Then by making slice for the action, one gets the result.

Now dP_{+} splits as:

dP_{+} = d_{1}P_{+} ⊕ d_{2}P_{+} : (W_{l}^{2})_{w}(M̂, g) ⊕ 𝔠 ↦ (W_{l−1}^{2})_{w}(M̂, g),

where d_{1}P_{+}(α) = P_{+}(g)(φ^{−1})^{∗}d_{A}(α)|(u, φ), and d_{2}P_{+}(r) = P_{+}(g)((φ^{−1})^{∗}(r^{∗}F)). We show that the differential of P_{+} is surjective.

Let us consider the AHS complex:

0 ⟶ (W_{l+1}^{2})_{w}(M̂, g) ⟶^{d_{A}} (W_{l}^{2})_{w}(M̂, g) ⟶^{d_{A}^{+}} (W_{l−1}^{2})_{w}(M̂, g) ⟶ 0.

Since this is Fredholm, one sees dP_{+} has finite codimension. Let us take a representative u ∈ CokerdP_{+} with (d_{A}^{+})_{w}^{∗}(u) = 0. Then one has d_{A}(e^{w}u) = 0. Then one has the equations, d_{A}(F_{A}) = d_{A}^{∗}(F_{A}) = d_{A}(v) = d_{A}^{∗}(v) = 0, where we use g′ = φ^{∗}(g) metric and v = φ^{∗}(e^{w}u) (here one uses Hilbert space structure of the function space, (W_{l−1}^{2})_{w}(M̂, g′; Λ_{+}^{2})). Then the same argument as [FU] p 56, shows that on open dense subset of M̂, F_{A} can be expressed as α ⊗ a ∈ Λ_{+}^{2}⊗Ad(E), with |a| = 1 (pointwise norm) and d_{A}(a) = 0. By the irreducibility, it follows a = 0, which contradicts to non triviality of p_{1}(E). Thus dP_{+}|(A, φ) is surjective. This completes the proof.

Corollary 6.1 For a Baire set of ϕ ∈ ℭ, there are no reducible connection in 𝔄(M̂) with respect to ϕ^{∗}(g).

P roof : Suppose A_{0} is reducible, and denote the corresponding U(1) connection by the same A_{0}. Let P be a G = U(1) bundle. Then as before one puts:

𝔄(P) = {A_{0} + a|a ∈ (W_{l}^{2})_{w}(M̂, g)},

\begin{matrix}𝔊(M̂)_{0} = \{h ∈ C_{loc}^{0}(Y; Aut(E))| & ∇_{A_{0}}h ∈ (W_{l}^{2})_{w}(M̂, g′), \\ |h − id|C^{0}(S^{3} × [i, i + 1]) → 0\}. \end{matrix}

Then 𝔊(M̂)_{0} acts on 𝔄(M̂). One considers P_{+} : 𝔄(M̂) × ℭ ↦ (W_{l−1}^{2})_{w}(M̂, g). dP_{+} is surjective at P_{+}(A, φ) = 0, since F_{A} ≠ 0.

Now one puts 𝔐̄(M̂) = P_{+}^{−1}(0)/𝔊(M̂)_{0}. Then one has a smooth Banach manifold 𝔐̄(M̂)′ = 𝔐̄(M̂)/𝔊(M̂)_{0}. Let us consider a Fredholm map between Banach manifolds π̄ : 𝔐̄(M̂)′ ↦ ℭ. Its Fredholm index is the one of the following one:

0 ⟶ (W_{l+1}^{2})_{w}(M̂, g) ⟶^{d} (W_{l}^{2})_{w}(M̂, g) ⟶^{d^{+}} (W_{l−1}^{2})_{w}(M̂, g) ⟶ 0.

one has H^{0}(AHS) = H^{1}(AHS) = 0, and H^{2}(AHS) = b_{+}^{2}(M). This is also the case when one purturbes the Riemannian metrics slightly. In particular, if b_{+}^{2} > 0, then the index is negative. Then the Sard Smale theorem shows that for a Baire set of ℭ, there are no ASD connections over non trivial line bundles. This shows that there are no orbits of reducible ASD connections in 𝔐(M̂) for a Baire set of ℭ. This completes the proof.

Let us put the projection as π : 𝔐̄(M̂)/𝔊(M̂) ↦ ℭ. Then the direct application of [FU] to this case, shows:

Corollary 6.2 (FU) Suppose b_{+}^{2} > 0. Then a Baire set of ϕ ∈ ℭ exists such that 𝔐̂(ϕ) ≡ π^{−1}(ϕ) are smooth finite dimensional manifolds.

6.C L_{w}^{p} spaces: let g′ be a small perturbation of g as above. Thus for any ASD connection A with respect to g′, Ker{d_{A} : 𝔤 = Lie𝔊 ↦ (W_{l}^{2})_{w}} = 0 and Cokerd_{A}^{+} : (W_{l}^{2})_{w} ↦ (W_{l−1}^{2})_{w} = 0. Let us recall that we have defined Banach spaces B^{∗}(M̂), B̄^{∗}(M̂) in 1.A, 2.A respectively (any element lies in L_{2w}^{4+2ϵ}(L̂) or L_{w}^{2+ϵ}(L̂)). Let A be a B̄^{1} ASD connection, A ∈ 𝔄(B̄^{∗}) = {A_{0} + a|a ∈ B̄^{1}(M̂; ad𝔤 ⊗ Λ^{1})}. One may assume A is smooth.

Lemma 6.3 For a Baire set of ϕ ∈ ℭ and g′ = ϕ^{∗}(g), the AHS complex:

0 ⟶ As𝔤(M̂, g′) ⟶^{d_{A}} B̄^{1}(M̂, g′; ad𝔤) ⟶^{d_{A}^{+}} B̄^{2}(M̂, g′; ad𝔤) ⟶ 0.

satisfies Kerd_{A} = 0 and Cokerd_{A}^{+} = 0.

P roof : When A is a B^{1}(M̂, 0) ASD connection, one may use the Hilbert space structure of B^{2}(ϵ = 0). Then one can follow a parallel argument as above, and one gets the conclusion that for a generic g′, there are no Kerd_{A} : As𝔤(ϵ = 0) ↦ B^{1}(ϵ = 0), and no Cokerd_{A}^{+} : B^{1}(ϵ = 0) ↦ B^{2}(ϵ = 0).

Let us use g′ above and take u ∈ Kerd_{A} : B̄^{0}(M̂, ϵ) ↦ B̄^{1}(M̂, ϵ). Since u satisfies the elliptic equation (d_{A})_{w}^{∗} ∘ d_{A}(u) = 0, u has the exponential decay (see the proof of sublemma 3.1). Thus there is a positive constant δ_{0} > 0 and for the corresponding weight function w_{0}, u lies in (W_{k}^{4+2ϵ})_{2w_{0}} for all k. Thus by the method in the proof of proposition 4.1, u ∈ (W_{l}^{4})_{2w}, 0 < δ < δ_{0}, which is a contradiction.

Next let us consider Cokerd_{A}^{+} : B̄^{1}(M̂, ϵ) ↦ B̄^{2}(M̂, ϵ). In 5.C, we have constructed Q_{A} : imd_{A}^{+} ∩ B̄^{2}(M̂, ϵ) ↦ B̄^{1}(M̂, ϵ) from Q_{A} : imd_{A}^{+} ∩ L_{w}^{2}(M̂) ↦ (W_{1}^{2})_{w}(M̂). In this case, imd_{A}^{+} = L_{w}^{2}(M̂). Thus one has Q_{A} : B̄^{2}(M̂, ϵ) ↦ B̄^{1}(M̂, ϵ) with d_{A}^{+} ∘ Q_{A} = id as desired. This completes the proof.

We want to have transversality for asymptotic Banach spaces B̂^{∗}({K_{l}}_{l}, ϵ). Let us construct a family of perturbed Riemannian metrics {g(l)}_{l}. Let g_{0} and g_{1} be cylindrical metric on M̂ and S^{3} × R respectively, and we take g′_{i}, small perturbations as above. Let g′_{i}(N) be another metrics with g′_{0}(N)|M̂\S^{3} × [N − 1, ∞) = g′_{0} and g′_{0}(N)|S^{3} × [N, ∞) = g_{0}. Similarly g′_{1}(N)|S^{3} × [−N + 1, ∞) = g′_{1} and g′_{1}(N)|S^{3} × (−∞, −N] = g_{1}. Then by regarding M̂ = M̂\S^{3} × [N, ∞) ∪ S^{3} × [−N, ∞), one gets another smooth Riemannian metric g(N) on M̂. This perturbation of Riemannian metrics works when one considers only one bubble at infinity. To treat asymptotic Banach spaces with n bubble, we perturb as follows. We recall a family of indices {a(i, j)} and translations T (i, j) i = 0, 1, …, j = 0, … n (2.A). We use the above notation. Then we choose Riemannian family g_{1}(i, j)′′ on S^{3} × [a(i, j − 1), a(i, j)] satisfying:

\begin{matrix}T (i, j)^{∗}\{g′′_{1}(i, j)|S^{3} × [a(i, j − 1) + 1, a(i, j) − 1]\} = g′_{1} \\ T (i, j)^{∗}\{g′′_{1}(i, j)|S^{3} × \{a(i, j − 1) ∪ a(i, j)\}\} = g_{1}. \end{matrix}

Then we identify M̂ = M̂\S^{3} × [a(i, 0), ∞) ∪ S^{3} × [a(i, 0), a(i, 1)] ∪ … S^{3} × [a(i, n − 1), a(i, n)] ∪ S^{3} × [a(i, n), ∞). Correspondingly we make a family of Riemannian metrics on M̂ as g(l) = g′_{0}(a(l, 0)) ∪ g′′_{1}(l, 1) ∪ ⋯ ∪ g′′(l, n).

So far we have fixed one Riemannian metric g, in particular when defineing asymptotic Banach spaces. Here we use a family of Riemannian metrics {g(l)}_{l}. Thus we formulate Banach spaces B̄^{∗}({K_{l}}_{l}, ϵ), using norms ||u_{l}||_{l} by the following:

|[u_{l}]|B̄^{∗}({K_{l}}_{l}; n) = Sup_{l}||u_{l}||_{l}
≡ Sup_{l}|φ_{a(l,0)}u_{l}|B̄^{∗}(K_{l}, w, ϵ, g′_{0}(a(l, 0)))
+ Σ_{j=1}^{n}|[φ_{a(l,j)} − φ_{a(l,j−1)}]u_{l}|B̄^{∗}(K_{l}, w(l, j), ϵ, g′′_{1}(l, j)).

Clearly these Banach spaces B̂^{∗}({K_{l}, g(l)}, ϵ) are equivalent to the previous B̂^{∗}({K_{l}}_{l}, ϵ).

Let us take A = [A_{l}] ∈ As𝔐({K_{l}, g(l)}_{l}, ϵ). We recall corollary 5.2, that there is a natural surjection:

As𝔐({K_{l}}_{l}, ϵ) ↦ ∪𝔐(M̂, g′_{0}) × ×_{n}𝔐(S^{3} × R, g′_{1})

by letting [A_{l}] → lim_{l}A_{l} = (A_{0}, …, A_{n}). Let us consider the AHS complex:

0 ⟶ As𝔤({K_{l}, g(l)}_{l}, ϵ) ⟶^{d_{A}} B̂^{1}({K_{l}, g(l)}_{l}, ϵ) ⟶^{d_{A}^{+}} B̂^{2}({K_{l}, g(l)}_{l}, ϵ) ⟶ 0.

We know that the above complex has closed range. Then we show Kerd_{A} = 0 and Cokerd_{A}^{+} = 0 as follows. Suppose A is reducible. Then there is [u_{l}] ∈ As𝔤 = LieAs𝔊 with d_{A}([u_{l}]) = 0 ∈ B̂^{1}(M̂). This shows lim_{l}||d_{A_{l}}(u_{l})||_{l}B̄^{1} = 0. By choosing sufficiently large l, this contradicts irreducibility of A_{j} over M̂ or S^{3} × R.

Next we show H_{A}^{2} = 0. By 5.B, one may choose [A′_{l}], As𝔊 gauge equivalent to A = [A_{l}] such that lim_{l}A_{l} = (A_{0}, …, A_{n}), where A_{j} are smooth ASD connections over M̂ or S^{3} × R. For every ASD connection A_{j}, there exists:

\begin{matrix}Q_{A_{0}} : B̄^{2}(M̂; Λ_{+}^{2} ⊗ ad𝔤) ↦ B̄^{1}(M̂; Λ^{1} ⊗ ad𝔤), \\ Q_{A_{j}} : B̄^{2}(S^{3} × R; Λ_{+}^{2} ⊗ ad𝔤) ↦ B̄^{1}(S^{3} × R; Λ^{1} ⊗ ad𝔤) \end{matrix}

with d_{A_{j}}^{+} ∘ Q_{A_{j}} is the identity. Then a parallel argument as the proof of lemma 3.1 shows that there exists Q_{A} : B̂^{2} ↦ B̂^{1} with d_{A}^{+} ∘ Q_{A} = id. In particular d_{[A′_{l}]}^{+} : B̂^{1}(M̂) ↦ B̂^{2}(M̂) is surjective. This completes the proof.

6.D Perturbation of asymptotic equivalence: Let K_{0} ⊂ K_{1} ⊂ ⋯ ⊂ M̂ be an exaustion, and take an asymptotic morphism {I_{l}}_{l} from (M̂, g) to (M̂′, g′). The morphism is defined using Riemannian metrics, and is not stable under the above perturbation. In order to overcome this, we will use more restrictive perturbation. Let us define:

ℭ = {Φ ∈ Diff_{0}^{l}(M̂) : inf_{Φ_{t}} Sup_{t}[|e^{h}[ id−Φ_{t}]|C^{0}+Σ_{j=1}^{l}|e^{h}∇^{j}( id−Φ_{t})|C^{0}] < ∞}

where we take all the family of diffeomorphisms Φ_{t} with ϕ_{0} = id and Φ_{1} = Φ. Thus we have a norm on T ℭ defined by ||Φ|| = inf_{Φ_{t}} Sup_{t}[|e^{h}Φ_{t}|C^{0} + Σ_{j=1}^{l}|e^{h}∇^{j}Φ_{t}|C^{0}].

Since M̂ is of bounded geometry, there is a small constant ϵ > 0 such that ℭ(ϵ) = {Φ ∈ ℭ : || id − Φ|| ≤ ϵ} is a Banach manifold. Then by a parallel argument, one gets the same conclusion as corollary 6.2 for ℭ(ϵ) by taking dΦ ∈ C^{l}(Gl(T M̂)).

Let us choose any Φ ∈ ℭ(M̂, ϵ) and Φ′ ∈ ℭ(M̂′, ϵ). Let us put:

h = (dΦ)^{∗}(g), h′ = (dΦ′)^{∗}(g′), J_{l} = (Φ′)^{−1} ∘ I_{l} ∘ Φ.

Then to give another asymptotic morphism, it is immediate to check the five defining conditions in the introduction for the following triples:

{J_{l}}_{l} : (M̂, h) ↦ (M̂′, h′).

When we perturb asymptotic Banach spaces as above, we choose Φ^{0} ∈ ℭ(M̂, ϵ), Φ^{1} ∈ ℭ(S^{3} × R, ϵ) and choose paths of diffeomorphisms Φ_{t}^{0} and Φ_{t}^{1}. Then by a similar method as 6.C, one obtains a family of diffeomorphisms {Φ(l)}_{l}. By choosing h(l) = dΦ(l)^{∗}(g) and J_{l} = (Φ(l)′)^{−1}I_{l}Φ(l) as above, one gets asymptotic Banach spaces {B̂^{∗}({K_{l}, Φ(l)}_{l}, ϵ)}_{∗}. The same consideration as 6.C, performs transversality for the AHS complex {B̂^{∗}({K_{l}, Φ(l)}_{l}, ϵ), d_{A}^{∗}}_{∗}.

7 Isomorphisms between moduli spaces

Let us take an asymptotic q.c. equivalence {I_{l}}_{l} with uniform bound, from (M̂, g) to (M̂′, g′). Let us choose small constants ϵ > ϵ′ > ϵ′′. Then by proposition 4.1, one gets a continuous map:

\begin{matrix}\{I_{l}^{∗}\}_{l} : (As𝔐(M̂, g, ϵ), 𝔐(M̂, g, ϵ)) ↦ (As𝔐(M̂′, g′, ϵ′), 𝔐(M̂′, g′, ϵ′)) \\ \{(I_{l}^{−1})^{∗}\}_{l} : (As𝔐(M̂′, g′, ϵ′), 𝔐(M̂′, g′, ϵ′)) ↦ (As𝔐(M̂, g, ϵ′′), 𝔐(M̂, g, ϵ′′)). \end{matrix}

One has seen that the composition of the above maps gives the identity. By symmetry, one finds that As𝔐(M̂, g) and As𝔐(M̂′, g′) are homeomorphic.

Proof of theorem 0.2: Suppose {I_{l}}_{l} consists of locally bounded as. q.c. equivalence, such that for any [A_{l}] ∈ As𝔐(M̂(′), g(′)),

lim_{r→0}lim sup_{x→∞} sup_{k,l}|I_{l}(^{−1})^{∗}(F_{A_{k}})|L^{2}(B_{r}(x)) = 0.

Then we show that for each pair (a, T), there is another one (a′, T′) such that {I_{l}}_{l} induces a homeomorphism:

{I_{l}^{∗}}_{l} : As𝔐(M̂, g; ϵ; a, T) ≅ As𝔐(M̂′, g′; ϵ′; a′, T′), ϵ′ < ϵ

for all metrics sufficiently near the cylindrical ones. By the construction, this assignment varies continuously with respect to the pair. Thus this will give the desired homeomorphism between parametrized asymptotic ASD moduli spaces.

Let us take any [A_{l}] ∈ As𝔐(M̂, g, ϵ). Passing through the surjection As𝔐 ↦ ∪𝔐(M̂)××_{n}𝔐(S^{3} ×R), [A_{l}] splits as A_{0} ××_{j=1}^{n}A_{j}. Let us take two exaustions by compact subsets on M̂ and S^{3} × R respectively. We denote them by the same K_{l} ⊂ M̂ or S^{3} × R. We consider the restriction I_{l}|K_{l_{0}} for l_{0} << l. Recall the family of translations {T (l, j)}. When K_{l_{0}} ⊂ S^{3} × R we identify I_{l}|K_{l_{0}} with I_{l}T (l_{0}, j)|K_{l_{0}}.

For every l_{0} << l, there exists a gauge transformation g_{l_{0}}^{j} ∈ B̄_{loc}^{0}(Aut(E|K′_{l_{0}})) with:

|(g_{l_{0}}^{j})^{∗}(I_{l}^{∗}(A_{j}|K_{l_{0}}))|B̄^{1}(K′_{l_{0}}, ϵ′)_{loc} ≤ C|I_{l}^{∗}(F_{A_{j}|K_{l_{0}}})|L̂^{2}(K′_{l_{0}}, ϵ′)_{loc}. ∗
|(g_{l_{0}}^{j})^{∗}(I_{l}^{∗}(A_{j}|K_{l_{0}}))|(W_{1}^{2+ϵ})_{loc}(K′_{l_{0}}) ≤ C|I_{l}^{∗}(F_{A_{j}|K_{l_{0}}})|L_{loc}^{2}(K′_{l_{0}}).

Recall that there is a family of measurable bundle maps τ_{l} ∈ L^{n}(K_{l}) with |τ_{l}|L^{n} → 0. Let us put A_{j}(l_{0}, l) ≡ (g_{l_{0}}^{j})^{∗}(I_{l}^{∗}(A_{j}|K_{l_{0}})) = d + a_{j}(l_{0}, l).

Sublemma 7.1 ([DS] p 244) A subsequence of {A_{j}(l_{0}, l)}_{l} converges to a g′ ASD connection in B̄^{1}(K′_{l_{0}}, ϵ′)_{loc} as l → ∞ for every l_{0}.

In fact by the condition |∇(I_{l} − I_{l′})|L_{loc}^{4+δ} → 0, one does not need to take subsequences.

Let us consider j = 0. Then from the sublemma, we have the following data:

(1) pair of subindices {k_{l}}_{l}, {m_{l}}_{l}, l << k_{l} << m_{l},

(2) another indices {a′(l, 0)}_{l} and translations {T′(l, 0)}_{l}

such that these data gives an element in

[A_{0}(l, m_{l})] ∈ As𝔐({K′_{k_{l}}, a′(l, 0), T′(l, 0)}_{l}; ϵ′).

One can check that this element does not depend on choice of {k_{l}}_{l} and {m_{l}}_{l}. Moreover a′ and T′ are essentially determined by [A_{l}].

For j ≥ 1, the situation is similar. Let j = 1 and put I(l_{0}, l) ≡ I_{l}T (l_{0}, 1)|K_{l_{0}}. Recall the exaustion K_{0} ⊂ K_{1} ⊂ ⋯ ⊂ S^{3}×R. Again in this case also, there are data as above with [A_{1}(l, m(l, 1))] ∈ As𝔐({K′_{k(l,1)}, a′(l, 1), T′(l, 1)}_{l}; ϵ′). One may assume {k(l, 1)}_{l} is a subindex of {k_{l}}_{l}. For j ≥ 2, we get also {k(l, j)}, {m(l, j)}_{l}, {a′(l, j)} and {T′(l, j)}. Let us put a′ = ∪_{j=0}^{n}{a′(l, j)}_{l} and T′ = ∪_{j=0}^{n}{T′(l, j)}_{l}. By gathering and taking subindices, one gets a total element:

I(A) = [Σ_{j=0}^{n}A′(l, j)] ∈ As𝔐({K′_{k(l,n)}, a′, T′}_{l}; ϵ′).

Thus for A = [A_{l}], we have chosen {k_{l}}_{l}, a′ and T′, and made another I(A). As above I(A) is independent of choice of {k_{l}}, and a′, and T′ are essentially deteremined by A. If another A′ ∈ As𝔐(M̂, g, ϵ) is sufficiently near A, then one may choose the same {k_{l}}_{l} as A, and the corresponding a′(A′) and T′(A′) are sufficiently near a′ = a′(A) and T = T′(A). Thus by taking countable dense subset of As𝔐(M̂, g, ϵ), one can choose the same {k_{l}}_{l} for all A ∈ As𝔐(M̂, g, ϵ) by the diagonal method. Moreover the corresponding a′ and T′ vary continuously with respect to A.

Now we have assigned [A′_{l}] ∈ As𝔐(M̂′, g′, ϵ′) to [A_{l}] ∈ As𝔐(M̂, g, ϵ). We show that this is well defined. We have chosen g_{l}^{j} ∈ C_{loc}^{0}(Aut(E|K′_{l})) to get A_{j}(l). Suppose another family {h_{l}^{j}}_{l} satisfies the same role as {g_{l}^{j}}_{l}. Let us put g′_{l} = (h_{l}^{j})^{−1}g_{l}^{j}, A_{j}(l) = (g_{l}^{j})^{∗}(I_{l}^{∗}(A_{j})) = d + a_{j}(l) and A′_{j}(l) = (h_{l}^{j})^{∗}(I_{l}^{∗}(A_{j})) = d+a′_{j}(l). For the moment we omit to denote j. Notice the equality (g′_{l})^{∗}(A′(l)) = (g′_{l})^{∗}(h_{l}^{∗}(I_{l}^{∗}(A))) = A(l). First we have an estimate:

||d(g′_{l})||L̂_{2w}^{4} ≤ C||(g′_{l})^{∗}(A′(l))||L̂_{2w}^{4}+C||a′_{j}(l)||L̂_{2w}^{4} = C{||a_{j}(l)||L̂_{2w}^{4}+||a′_{j}(l)||L̂_{2w}^{4}}.

In particular ||g′_{l}||B̄^{0} are uniformly bounded. Then by taking a subsequence, one may assume {g′_{l}}_{l} converges in C^{0} topology. Then we have estimates:

||d(g′_{l} − g′_{l′})||L̂_{2w}^{4} ≤ C{||(g′_{l} − g′_{l′})^{∗}[(h_{l}^{∗}(I_{l}^{∗}(A)))]||L̂_{2w}^{4} + ||(g′_{l} − g′_{l′})^{∗}(a′_{j}(l))||L̂_{2w}^{4}}
≤ C{||(g′_{l})^{∗}(A′(l)) − (g′_{l′})^{∗}(A′(l′))||L̂_{2w}^{4}
+ ||(g′_{l′})^{∗}[A′(l) − A′(l′)]||L̂_{2w}^{4} + |g′_{l} − g′_{l′}|C^{0}||a′_{j}(l)||L̂_{2w}^{4}}

≤ C{||A(l) − A(l′)||L̂_{2w}^{4} + ||A′(l) − A′(l′)]||L̂_{2w}^{4} + |g′_{l} − g′_{l′}|C^{0}||a′_{j}(l)||L̂_{2w}^{4}}.

Thus by the above estimate, one sees [g′_{l}] ∈ As𝔊(M̂′, g′, ϵ′). Thus we have verified that the above assignment is independent of choice of local gauge transformations g_{l}^{j}.

Next suppose [A_{l}], [A′_{l}] ∈ As𝔐(M̂, g, ϵ) are As𝔊(M̂, g, ϵ) equivalent by [g_{l}]. Let us choose [h_{l}] and [h′_{l}] for [A_{l}] and [A′_{l}] so that one gets [A(l)], [A′(l)] ∈ As𝔐(M̂′, g′, ϵ′) as above. Let us put u_{l} = g_{l} ∘ I_{l}. Then clearly I_{l}^{∗}(A′_{l}) = I_{l}^{∗}(g_{l}^{∗}(A_{l})) = u_{l}^{∗}(I_{l}^{∗}(A_{l})). Thus we compare [h_{l}^{∗}(I_{l}^{∗}(A_{l}))] and [(h′_{l})^{∗}u_{l}^{∗}(I_{l}^{∗}(A_{l}))]. Then one sees [(h_{l}^{−1})(u_{l}h′_{l})] ∈ As𝔊 by the above argument. Thus [A(l)] and [A′(l)] are As𝔊(M̂′, g′, ϵ′) equivalent.

Now we have a well defined map I^{∗} : As𝔐(M̂, g, ϵ) ↦ As𝔐(M̂′, g′, ϵ′). By the same way, one gets another map (I^{−1})^{∗} : As𝔐(M̂′, g′, ϵ′) ↦ As𝔐(M̂, g, ϵ′′). Next we see that the composition of these is the identity. Let us take [A_{l}] ∈ As𝔐(M̂, g, ϵ). The composition is given as [g_{l}^{∗}(A_{l})] ∈ As𝔐(M̂, g, ϵ′′). Then by the same way as above, one sees [g_{l}] ∈ As𝔊(M̂, g, ϵ′′). This verifies that the composition (I^{−1})^{∗} ∘ I^{∗} gives the identity. One knows also the converse composition I^{∗} ∘(I^{−1})^{∗} = id. Thus combining with corollary 5.1, this completes the proof of theorem 0.2.

8 Large deformation of ASD moduli spaces

Let M be a closed C^{∞} oriented four manifold, and for some R ⊂ M as in the introduction, one gets M̌ = M\R̄ ⊂ M and M̂ = M\ pt. Thus M̌ and M̂ are smooth submanifolds of M. Suppose M̌ is asymptotically quasiconformally homeomorphic to (M̂, g). Let us take an ACH : M̌ → (M̂, g), {I_{l} : (K_{l}, g_{l}) ↦ (M̂, g)}_{l},

Let us take a generic family of Riemannian metrics h_{l} on M with |g_{l} − h_{l}|K_{l}|C^{∞} → 0. Notice that by changing g_{l} by h_{l}, the data I_{l} : (K_{l}, h_{l}) ↪ (M̂, g) satisfies the defining conditions for ACH. Let us take an SO(3) bundle E → M, and construct a family of ASD moduli spaces {𝔐(M, h_{l})}_{l}. Suppose they are all non-empty, and take any elements [A_{l}] ∈ 𝔐(M, h_{l}). Here we will use Banach spaces B(M̂, ϵ) in 1.A. Recall that for an SO(3) bundle E′ → M̂ with fixed trivialization on the end, one has a formal affine Banach space 𝔄 = {A_{0} + a|a ∈ B^{1}(M̂, ϵ)} where A_{0} is a smooth compactly supported connection. Using 𝔄, one gets the ASD moduli space 𝔐(M̂, ϵ).

Proposition 8.1 After gauge transformation, a subsequence {A′_{l} = (I_{l}^{−1})^{∗}(A_{l})}_{l} converges to an element A_{∞} ∈ 𝔐(M̂, ϵ).

P roof : Here one may assume that all A_{l} are in C^{∞}. After gauge transformation and taking subsequence, one may assume (1) {A_{l}} converges on every compact subset and (2) one has a uniform estimate |A_{l}|(W_{k}^{p})_{loc} ≤ C|F_{A_{l}}|L_{loc}^{2} by taking a trivialization.

Let us consider a family {(I_{l}^{−1})^{∗}(A_{l})}_{l} over M̂. By lemma 4.1 (1), one gets the estimate:

|(I_{l}^{−1})^{∗}(A_{l}) − (I_{l′}^{−1})^{∗}(A_{l′})|L_{loc}^{4+2ϵ′}
≤ |(I_{l}^{−1})^{∗}(A_{l}) − (I_{l}^{−1})^{∗}(A_{l′})|L_{loc}^{4+2ϵ′} + |(I_{l}^{−1} − I_{l′}^{−1})^{∗}(A_{l′})|L_{loc}^{4+2ϵ′}

≤ C(|∇I_{l}^{−1}|L_{loc}^{4+δ})|A_{l} − A_{l′}|L_{loc}^{4+2ϵ} + C(|∇(I_{l}^{−1} − I_{l′}^{−1})|L_{loc}^{4+δ})|A_{l′}|L_{loc}^{4+2ϵ} → 0.

|(I_{l}^{−1})^{∗}(A_{l})|L_{loc}^{4+2ϵ′} ≤ C(|∇I_{l}^{−1}|L_{loc}^{4+δ})|A_{l}|L_{loc}^{4+2ϵ} ≤ C(|∇I_{l}^{−1}|L_{loc}^{4+δ})|F_{A_{l}}|L_{loc}^{2}.

Thus the family {(I_{l}^{−1})^{∗}(A_{l})}_{l} converges to A_{∞} in L_{loc}^{4+2ϵ′} topology. By considering similarly {F_{A′_{l}}}_{l} also converges in L_{loc}^{2+ϵ′}. Let ( )^{+} be the projection to the self dual part with respect to g. Then we have the equality F_{A′_{l}}^{+} + τ_{l}(F_{A′_{l}}) = 0, where |τ_{l}|L^{n} → 0 for large n. Combining with the estimate |τ_{l}F_{A′_{l}}|L^{2} ≤ C|τ_{l}|L^{n}|F_{A_{l}}|L^{2+ϵ}, n^{−1} + (2 + ϵ′)^{−1} = 1, we see F_{A_{∞}}^{+} = 0. Thus A_{∞} is g-ASD connection. After gauge transformation by h, one may assume h^{∗}(A_{∞}) satisfies the exponential decy estimate in W_{k}^{2} norm. Since F_{h^{∗}(A_{∞})} = h^{∗}(F_{A_{∞}}), the L^{2} norms are unchanged under gauge transformation. In particular F_{A_{∞}} satisfies exponential decay estimate in L^{2} norm.

Let us consider the following estimates:

|(I_{l}^{−1})^{∗}(A_{l})|L_{loc}^{4+2ϵ′} ≤ C|F_{A_{l}}|L_{loc}^{2}
= C|I_{l}^{∗}(F_{(I_{l}^{−1})^{∗}(A_{l})})|L_{loc}^{2} ≤ C′|F_{(I_{l}^{−1})^{∗}(A_{l})})|L_{loc}^{2},
|F_{(I_{l}^{−1})^{∗}(A_{l})}|L_{loc}^{2+ϵ′} = |(I_{l}^{−1})^{∗}(F_{A_{l}})|L_{loc}^{2+ϵ′} ≤ C|F_{A_{l}}|L_{loc}^{2+ϵ}
≤ C′|F_{A_{l}}|L_{loc}^{2} = C′|I_{l}^{∗}(F_{(I_{l}^{−1})^{∗}(A_{l})})|L_{loc}^{2} ≤ C′′|F_{(I_{l}^{−1})^{∗}(A_{l})}|L_{loc}^{2}.

¿From these estimate, one gets an element A_{∞} ∈ 𝔐(M̂, g, ϵ′). This completes the proof.

9 Donaldson’s invariant

Let M be an oriented simply connected smooth closed four manifold. Let h be any element in H_{2}(M; Z). One may represent h by a smoothly embedded Riemann surface. In fact by Hurewicz surjection, h has a representative by a smooth immersion f : S^{2} ↦ M. One may assume that the self intersection number of f is zero. For each Whitney pair of self intersection points (p_{0}, p_{1}), one has an embedded boundary of a Whitney disk l = l_{0} ∪ l_{1}, where l_{i} are arcs with two points l_{0} ∩ l_{1} = {p_{0}, p_{1}} as the Whitney pair. Remove small neighbourhoods B_{ϵ}(p_{i}), and add a small cylinder S^{1} ×[0, 1] along ∂B_{ϵ}(p_{i}) which contains l_{0} in the cylinder. The result is an embedded Riemann surface which represents the same homology class as h (notice that H_{2}(M; Z) is torsion free).

Let Σ_{1}, …, Σ_{k} ⊂ M be embedded surfaces representing homology classes [Σ_{i}] ∈ H_{2}(M; Z). Then one can construct Q(E; M, h_{1}, …, h_{m}), the Donaldson’s invariant We follow [FS2]. Let E be an SO(3) bundle over M, and take any embedded Riemann surface Σ ⊂ M. Let 𝔅_{k}^{∗}(Σ; E|Σ) = 𝔄(E|Σ)_{k}^{∗}/𝔊_{k+1}(E|Σ) be the gauge equivalence classes of irreducible Sobolev k connections. Similarly one has 𝔅̃_{k}^{∗}(Σ; E|Σ) = 𝔄(E|Σ)_{k}^{∗}/𝔊_{k+1}(E|Σ)_{0}, where 𝔊_{k+1}(E|Σ)_{0} ⊂ 𝔊_{k+1}(E|Σ) is a subgroup with g(x_{0}) is the identity for a fixed point x_{0} ∈ Σ. Thus one has a fibration:

SO(3) ↪ 𝔅̃_{k}^{∗}(Σ; E|Σ) ↦ 𝔅_{k}^{∗}(Σ; E|Σ)

which gives an SO(3) principal bundle β over 𝔅_{k}^{∗}(Σ; E|Σ). Now one has the following:

Sublemma 9.1 (FS2) (1) If < w_{2}(E), Σ >≠ 0, then β lifts to a U(2) bundle β̃.

(2) For Σ, there is another embedded surface Σ′ ⊂ M such that restriction of any irreducible connection A over E to E|Σ is also irreducible.

For our application, it is enough to assume the condition (1) in the above sublemma. Thus let us choose a lift β̃ over 𝔅_{k}^{∗}(Σ; E|Σ). One has c_{1}(β̃) = w_{2}(β) mod 2. Let us choose a complex line bundle Λ over 𝔅_{k}^{∗}(Σ; E|Σ) with c_{1}(Λ) = c_{1}(β̃).

By the above sublemma (2), one has the restriction map:

r : 𝔐_{k}^{∗}(E) ↦ 𝔅_{k}^{∗}(Σ; E|Σ).

One chooses a generic section s of Λ. Then by the transversality argument ([DK] p192), one has a smooth manifold 𝔐_{k}^{∗}(E) ∩ r^{−1}( Kers) of codimension 2. One may generalize this construction. Suppose dim𝔐_{∗}(E) = 2l, and choose l embedded surfaces Σ_{1}, …, Σ_{l} ⊂ M. As above one has determinant bundles over tublar neighbourhoods of Σ_{i}, ν(Σ_{i}). Then for generic sections, s_{1}, …, s_{l}, one has a smooth submanifold of dimension 0 as:

𝔐_{∗}(E) ∩ r_{1}^{−1}( Kers_{1}) ∩ ⋯ ∩ r_{l}^{−1}( Kers_{l}).

By choosing these embeddings with general positions, one may assume that this 0 dimensional manifold consists of finite oriented points. Thus one may count the number of points so that one obtains an integer. This number is, by definition, Q(M, E; [Σ_{1}], …, [Σ_{l}]) (modulo constant).

10 Proof of theorem 1

Let X be a simply connected smooth four manifold with even type form (H_{2}(X : Z), σ) ≅ (H_{2}(k| − E_{8}| : Z) ⊕ H_{2}(l(S^{2} × S^{2}) : Z), −kE_{8} ⊕ lH), where σ is the form of X and H is the form of S^{2} × S^{2}. We take an SO(3) bundle E → X. Let w_{2}(E) = α̂ = α̂_{1} ⊕ α̂_{2} and take [Σ_{i}] = h_{i}^{1} ⊕ h_{i}^{2} ∈ H_{2}(X : Z). Then we get the following:

Theorem 10.1 Suppose all h_{i}^{1} = 0, 1 ≤ i ≤ l and α̂_{i} ≠ 0, i = 1, 2. Then there exists a pair (M, R) with the following property; suppose M̌ admits an asymptotically conformal equivalence to (M̂, g). Then the Donaldson’s invariant vanishes Q(E, X; [Σ_{1}], …, [Σ_{l}]) = 0.

P roof : Let us fix a marking of H^{2}(X; Z) as above, and take a smooth embedding M̌ = l(S^{2} × S^{2})\R̄ ↪ X, where R̄ is a topologically closed disk, and this embedding is compatible with the forms. By taking any integral lift α ∈ H^{2}(X; Z), it gives a lift of the SO(3) bundle on another U(2) bundle Ē.

Then for any h ∈ H_{2}(X : Z) and its mod 2 reduction [h]_{2}, one has the equality < w_{2}(E), [h]_{2} >=< c_{1}(Ē), h > mod 2.

M̌ is an open manifold on which one can equip an asymptotically quasiconformal homeomorphism to (M̂, g). Let us choose any complete Riemannian metrics of bounded geometry h on the interiors of N = X\M̌.

Let Σ_{1}, …, Σ_{m′} ⊂ M̂ be embedded surfaces as above and take another SO(3) bundle E′ → M̂. Then one has a smooth moduli space 𝔐(M̂, E′) of B^{1} ASD connections over E′ of dimension 2m, and 𝔐(E′, M̂; [Σ_{1}], …, [Σ_{m′}]) defined similarly as before. This space has a formal dimension 2(m − m′).

Now let Σ_{1}, …, Σ_{m} ⊂ M̌ ⊂ X be as the theorem. One may assume the general position of these; for any Σ_{i}, Σ_{j}, Σ_{k}, one has empty intersection Σ_{i} ∩ Σ_{j} ∩ Σ_{k} = ϕ. Suppose one has Q(E, X; [Σ_{1}], …, [Σ_{m}]) ≠ 0. This shows that for a generic choice of Riemannian metric g on X, and also generic choice of these surfaces, one has non empty 0 dimensional moduli space 𝔐(E, X; [Σ_{1}], …, [Σ_{m}]) whose algebraic number is equal to the above invariant Q.

Let us choose exausting families of compact subsets as:

K_{0} ⊂ K_{1} ⊂ ⋯ ⊂ M̌, K′_{0} ⊂ K′_{1} ⊂ ⋯ ⊂ N.

Let us choose a family of smooth Riemannian metrics g′_{l} on X with (I_{l}^{−1})^{∗}g′_{l} ∼ g, g′_{l}|K′_{l} = h|K′_{l} as in the definition of ACH, and choose a family of small constants δ_{i} > 0, i = 0, 1, …, with δ_{i} → 0. In order to construct a smooth moduli space 𝔐(E, X; [Σ_{1}], …, [Σ_{m}]), one needs to perturbe g′_{l} slightly. Let us choose a family of generic Riemannian metrics g_{l} on X with |g_{l} − g′_{l}|W_{k}^{2}(K_{i} ∪ K′_{i}) ≤ δ_{l}. Then one has a family of non empty smooth manifolds:

{𝔐(E, (X, g_{i}); [Σ_{1}], …, [Σ_{m}])}_{i≥0}.

Let us choose any sequence [A_{i}] ∈ 𝔐(E, (X, g_{i}); [Σ_{1}], …, [Σ_{m}]), i = 0, 1, . . .. Then there is a subsequence which we also denote by [A_{i}] so that (I_{l}^{−1})^{∗}(A_{l}) converges to [A_{∞}] ∈ 𝔐(M̂, ϵ′), an ASD connection by proposition 8.1. On the other hand, A_{l}|N converges to a h−ASD connection A′_{∞} over (N, h) with F_{A′_{∞}}|L^{2} < ∞.

Now let {p_{1}, …, p_{l}} ⊂ M̌ be the bubbling points. Then at most 2l surfaces can contain these points. Thus by reordering surfaces, one has:

A_{∞} ∈ 𝔐(E′, M̂, ϵ′; [Σ_{1}], …, [Σ_{m−2l}]).

Let us see that A_{∞} defines an SO(3) bundle E′ on M̂ with strictly smaller |p_{1}(E′)| than |p_{1}(E)|. For this, one will see |F_{A′_{∞}}|L^{2}(N, h) > 0. One has chosen E so that each α̂_{j} ≠ 0 where w_{2}(E) = α̂ = Σ_{j=1}^{2}α̂_{j} ∈ ⊕_{k}Z_{2}^{8}⊕⊕_{2l}Z_{2} ≅ H^{2}(X; Z_{2}).

Let α_{j} ∈ H^{2}(X; Z) be the corresponding lift. Let us take homology classes h = h_{1} + h_{2} ∈ H_{2}(X; Z) such that the mod 2 reduction of < α_{j}, h_{j} >_{2}∈ Z_{2} are all non zero (= 1). Notice w_{2}(E′) = α̂_{1} ∈ H^{2}(M̂; Z_{2}). Let Σ_{j} ⊂ X be embedded surfaces which represent h_{j}. E|Σ_{1} is isomorphic to E′|Σ_{1}. In particular one has w_{2}(E′) ≠ 0. If the family {F_{A_{i}}}_{i} concentrates at some points p ∈ N, then clearly one has |p_{1}(E′)| ≤ |p_{1}(E)| − 4. Now suppose it does not concentrate on N. Let Σ_{2} ⊂ N as above. Then one has the equality < w_{2}(E), Σ_{2} >=< c_{1}(Ē), Σ_{2} >=< c_{1}(A_{i}), Σ_{2} >≠ 0 mod 2. Since there are no concentrating points, one has also < c_{1}(A′_{∞}), Σ_{2} >≠ 0. Then clearly lim sup_{i}|F_{A_{i}}|L^{2}(N, g_{i}|N) ≥ 4π. Thus in any situation, one has |p_{1}(E′)| ≤ |p_{1}(E)|−2.

Now one has 𝔐(E, M̂; [Σ_{1}], …, [Σ_{m−2l}]), non empty moduli space of ASD connections. One may assume that this space contains no reducible orbits. Recall dim𝔐 ≤ dim Kerd_{A}^{+}/ imd_{A} − 4 − 2m, while dim Kerd_{A}^{+}/ imd_{A} − 2m = 0. Thus the formal dimension of this space is negative. This is a contradiction. This completes the proof of the theorem.

Proof of theorem 0.1: First we will find some X, an SO(3) bundle E → X and some embedded surface Σ ⊂ X with non zero Donaldson’s invariant Q(X, E, [Σ], …, [Σ]) ≠ 0. Donaldson verified that one finds such pair (X, E) if X is a simply connected algebraic surface, |p_{1}(E)| is sufficiently large and w_{2}(E) is mod 2 reduction of 1, 1 form.

Next one can choose a particular marking (H_{2}(X : Z), σ) ≅ (H_{2}(k| − E_{8}| : Z)⊕H_{2}(l(S^{2}×S^{2}) : Z), −kE_{8}⊕lH) so that for the corresponding decomposition [Σ] = h_{1}+h_{2} and w_{2}(E) = α_{1}+α_{2}, h_{1} vanishes and α_{i} ≠ 0, if the Picard number of X is sufficiently large. This follows from the existence of automorphisms of even type lattices. For example projective Kummer surface satisfies this condition (see [K] section 3).

The combination of these two steps gives the proof.

11 Asymptotic ASD moduli spaces

11.A Proof of corollary 0.1: Recall the statement:

Theorem 11.1 If the Donaldson’s invariants of M and N are different from each other, then M̂ and N̂ cannot be equivalent.

For simplicity we assume that 𝔐(M̂) and 𝔐(N̂) are generic ASD moduli spaces of dimension 0. Then the algebraic numbers of them are diffeomorphism type invariant. Now there are examples of M and N so that ♯𝔐(M̂) and ♯𝔐(N̂) are different. In particular they are non-diffeomorphic.

The proof is now at hand. Suppose contrary to the conclusion of the above theorem. Let {I_{l}}_{l} gives an equivalence and choose ϵ′ < ϵ. Then one gets homeomorphisms as:

{I_{l}^{∗}}_{l} : 𝔐(M̂, [Σ_{i}]) ≅ As𝔐(M̂, [Σ_{i}], ϵ) ↦ As𝔐(N̂, [Σ_{i}], ϵ′) ≅ 𝔐(N̂, [Σ_{i}]).

We have seen in section 7 that the middle map gives a homeomorphism. In particular ♯𝔐(M̂, [Σ_{i}]) = ♯𝔐(N̂, [Σ_{i}]). This is a contradiction.

11.B Asymptotic moduli spaces with low energy: Let {I_{l}}_{l} be an asymptotic morphism from (M̂, g) to (M̂′, g′).

Let us take isomorphic SO(3) bundles E and E′ over M̂ and M̂′ respectively with 12 ≥ −2p_{1}(E) − 3(1 + b_{2}^{+}(M)) = −2p_{1}(E′) − 3(1 + b_{2}^{+}(M)) > 0.

Theorem 11.2 Suppose the above sutuation. Then the induced morphism:

{I_{l}^{∗}}_{l} : P As𝔐(E, M̂, g, ϵ) ≅ P As𝔐(E′, M̂′, g′, ϵ′)

gives a homeomorphism.

P roof : Let us consider a family [A_{l} = I_{l}^{∗}(A)] where A ∈ As𝔐(M̂, ϵ). Consider a situation; for some sequences x_{0}, x_{1}, … and small constants μ_{0} >> μ_{1} >> ⋯ → 0, |F_{A_{l}}|L^{2}(B_{μ_{i}}(x_{i})) ≥ μ > 0. Notice that by the properties of I_{l}, x_{i} must go to infinity. However in our setting, such situation cannot occur, which follows from compactness of 𝔐(E′′, S^{3} × R, ϵ). This completes the proof.

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Tsuyoshi Kato Department of Mathematics Faculty of Science Kyoto University Kyoto 606-01 Japan